%------------------------------------------------------------------------------
% File     : ITP290_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DelImperative 00505_031086
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0095_VEBT_DelImperative_00505_031086 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 14799 (3844 unt;3545 typ;   0 def)
%            Number of atoms       : 30307 (10790 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 22197 (3144   ~; 547   |;2010   &)
%                                         (2637 <=>;13859  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   39 (   2 avg)
%            Number of types       :  819 ( 818 usr)
%            Number of type conns  : 3398 (2215   >;1183   *;   0   +;   0  <<)
%            Number of predicates  :  101 (  98 usr;   2 prp; 0-3 aty)
%            Number of functors    : 2629 (2629 usr; 512 con; 0-4 aty)
%            Number of variables   : 31759 (30814   !; 945   ?;31759   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 22:49:43.797
%------------------------------------------------------------------------------
% Could-be-implicit typings (818)
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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tff(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    set_VEBT_VEBT: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    fun_nat_complex: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
    fun_int_complex: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_J,type,
    fun_complex_rat: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Num__Onum_J,type,
    fun_complex_num: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
    fun_complex_int: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

tff(ty_n_t__Set__Oset_It__Numeral____Type__Onum1_J,type,
    set_Numeral_num1: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    fun_real_real: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    fun_real_bool: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Real__Oreal_J,type,
    fun_bool_real: $tType ).

tff(ty_n_t__itself_It__Numeral____Type__Onum1_J,type,
    itself_Numeral_num1: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Num__Onum_J,type,
    fun_real_num: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
    fun_real_nat: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    fun_real_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Real__Oreal_J,type,
    fun_rat_real: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Real__Oreal_J,type,
    fun_num_real: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    fun_nat_real: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_J,type,
    fun_int_real: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__Int__Oint_J,type,
    fun_bool_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    fun_rat_rat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Num__Onum_J,type,
    fun_rat_num: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
    fun_rat_nat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
    fun_rat_int: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
    fun_num_rat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    fun_num_num: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
    fun_num_nat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    fun_num_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
    fun_nat_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
    fun_int_num: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    fun_int_int: $tType ).

tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

tff(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
    option_real: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
    itself_finite_2: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____1_J,type,
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tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

tff(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_VEBTi: $tType ).

tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

tff(ty_n_t__Heap__Oarray_It__HOL__Obool_J,type,
    array_bool: $tType ).

tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
    list_bool: $tType ).

tff(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
    array_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
    list_num: $tType ).

tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

tff(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_n_t__Set__Oset_It__HOL__Obool_J,type,
    set_bool: $tType ).

tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

tff(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

tff(ty_n_t__Numeral____Type__Onum1,type,
    numeral_num1: $tType ).

tff(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

tff(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

tff(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

tff(ty_n_t__Uint32__Ouint32,type,
    uint32: $tType ).

tff(ty_n_t__Real__Oreal,type,
    real: $tType ).

tff(ty_n_t__HOL__Obool,type,
    bool: $tType ).

tff(ty_n_t__Rat__Orat,type,
    rat: $tType ).

tff(ty_n_t__Num__Onum,type,
    num: $tType ).

tff(ty_n_t__Nat__Onat,type,
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tff(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (2727)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_VEBT_VEBT_assn > fun_fu5192204193588358178T_assn ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_real_assn > fun_fu9107578059089610152l_assn ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_nat_assn > fun_fu6395640590628241448t_assn ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_int_assn > fun_fu4854284890850328872t_assn ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_Co4490034186337852034r_assn > fun_fu5114706524692223720r_assn ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_complex_assn > fun_fu7962712148269509288x_assn ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_nat_real > fun_fu9152424228179946668t_real ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: code_integer > fun_fu7011271208229444932nteger ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_int_assn > fun_int_bool ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_Co4490034186337852034r_assn > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_complex_assn > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: fun_Co3960896333048504454r_real > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_Code_integer_rat > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_VEBT_VEBT_real > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_VE5569070535936029429omplex > fun_fu5830828964019696513T_real ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_nat_Code_integer > fun_fu3510149322827840670nteger ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_nat_Code_integer > fun_fu7333075202678876331nteger ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_Pr958786334691620121nt_int > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: fun_nat_Code_integer > fun_Co3729874436117207851nteger ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: nat > fun_fu8719411193169181957nteger ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: fun_nat_Code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: nat > fun_fu8719411193169181957nteger ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__afu____,type,
    aTP_Lamm_afu: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__afv____,type,
    aTP_Lamm_afv: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_real_real ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: nat > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: nat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: nat > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: nat > fun_li5996000409811367950l_assn ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: nat > fun_li714799757257233060t_assn ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: nat > fun_li3423855071176226084i_assn ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__aha____,type,
    aTP_Lamm_aha: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: fun_real_bool ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: fun_real_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aif____,type,
    aTP_Lamm_aif: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aig____,type,
    aTP_Lamm_aig: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: fun_int_bool ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_int ).

tff(sy_c_ATP_058Lamm__aja____,type,
    aTP_Lamm_aja: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: fun_real_real ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ajr____,type,
    aTP_Lamm_ajr: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: fun_real_real ).

tff(sy_c_ATP_058Lamm__aju____,type,
    aTP_Lamm_aju: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__akm____,type,
    aTP_Lamm_akm: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: fun_real_real ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: fun_real_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__akz____,type,
    aTP_Lamm_akz: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_int_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_VEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_Pr958786334691620121nt_int > fun_fu7672510408924296255t_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_VEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_Pr958786334691620121nt_int > fun_se1375549587806057485t_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_real_real ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_int_int ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: vEBT_VEBT > fun_VE3808733824349961398l_assn ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: vEBT_VEBT > fun_VE4942641508680916543l_assn ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_bool_assn > fun_as6968651184343528707l_assn ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: vEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: product_prod_int_int > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: vEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: product_prod_int_int > fun_se1375549587806057485t_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: vEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: product_prod_int_int > fun_fu7672510408924296255t_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_VEBT_VEBT_bool > fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_complex_bool > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_Pr803113274115125473t_bool > fun_Pr3099820731930151597t_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_VEBT_VEBT_bool > fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_complex_bool > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_Pr803113274115125473t_bool > fun_Pr3099820731930151597t_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_VE1571388752190420990T_assn > fun_li4421246802284715015i_assn ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_VE4931415397359990460t_assn > fun_li5819835350756573403i_assn ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_VE1323945509527881313l_assn > fun_li4497802875955143402i_assn ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_VE6910866152645720856l_assn > fun_li6826900772859087287i_assn ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_re8050108611700907676T_assn > fun_li8670275784065541403i_assn ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_re560840401387337987i_assn > fun_li4242333044386785708i_assn ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_re574865750389657758t_assn > fun_li5176349751308624583i_assn ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_re2206215847901373695l_assn > fun_li7859777374269373438i_assn ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_re6303261851654292474l_assn > fun_li8943242453490873763i_assn ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_bo5394993412183903963T_assn > fun_li8745411898289106916i_assn ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_VE597728521808591204at_nat ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_bool_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_VE4436863797057549263at_nat ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: nat > fun_li6689927339917241139i_assn ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: nat > fun_li8221786654549472865l_assn ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: nat > fun_li7646410839599160030t_assn ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: nat > fun_li6911922195568332558t_assn ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: assn > fun_VE2561185598974572172i_assn ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: assn > fun_bo8172996160782635326l_assn ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: assn > fun_op6483474040057813218t_assn ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: assn > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_bool_assn > fun_as6968651184343528707l_assn ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: assn > fun_li3352006551604695530i_assn ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: assn > fun_fu5741754899130534993i_assn ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_li3352006551604695530i_assn > fun_as5323700489645809679i_assn ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_VE3708348504534315036i_assn > fun_li8825114373765265308i_assn ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_VE3708348504534315036i_assn > fun_VEBT_VEBTi_assn ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_bo5386306568165246356i_assn > fun_li4779488348070916006l_assn ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_bo5386306568165246356i_assn > fun_bool_assn ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_op8552747401449489031i_assn > fun_li3309467822378783623t_assn ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_op8552747401449489031i_assn > fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_na8005944528675429559i_assn > fun_li7023630697154767927t_assn ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_na8005944528675429559i_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_li8825114373765265308i_assn > fun_VE3708348504534315036i_assn ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_li8825114373765265308i_assn > fun_VEBT_VEBTi_assn ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_li4779488348070916006l_assn > fun_bo5386306568165246356i_assn ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_li4779488348070916006l_assn > fun_bool_assn ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_li3309467822378783623t_assn > fun_op8552747401449489031i_assn ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_li3309467822378783623t_assn > fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_li7023630697154767927t_assn > fun_na8005944528675429559i_assn ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_li7023630697154767927t_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: assn > fun_fu8979060164380311043i_assn ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: assn > fun_fu8850137702660867424i_assn ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: assn > fun_fu2678428474534703935l_assn ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: assn > fun_fu1723287682096855374l_assn ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: assn > fun_fu8734043002210220995t_assn ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: assn > fun_fu8831737957336725536t_assn ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: assn > fun_fu96848488177702211t_assn ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: assn > fun_fu1092876921774764064t_assn ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_int > fun_se6049520964208396421t_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_Code_integer > fun_se4340818509347730467r_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_complex > fun_se2925769915755310977x_bool ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_nat > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_Code_integer > fun_na3912021598713191116r_bool ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_VEBT_VEBTi > fun_na4319918184945908084i_bool ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_Code_integer > fun_na3912021598713191116r_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_VEBT_VEBTi > fun_na4319918184945908084i_bool ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: vEBT_VEBT > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: vEBT_VEBT > fun_bool_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: vEBT_VEBT > fun_VE9120655849212850072T_VEBT ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_Code_integer > fun_se2712834123562403997r_bool ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_Pr803113274115125473t_bool > fun_fu7672510408924296255t_bool ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_Co8822979462182099027r_bool > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_Pr803113274115125473t_bool > fun_fu7672510408924296255t_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Co8822979462182099027r_bool > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_VEBT_VEBT > fun_fu2721280555464079775T_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_VEBT_VEBT > fun_fu7178057608955062431T_bool ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_VEBT_VEBT > fun_fu7855050327546112735T_bool ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_real > fun_fu6700586743015562435l_bool ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_VEBT_VEBT > fun_fu748158482767045919T_bool ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_Pr5160692531401422620nt_nat > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_Pr5160692531401422620nt_nat > fun_fu600552898087764240t_bool ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_nat > fun_fu8422113094057456412t_bool ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_int_nat > fun_fu5706916984221235812t_bool ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_Code_integer_nat > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_Code_integer_nat > fun_fu7000754783045315842r_bool ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_complex_nat > fun_fu5450758583235971936x_bool ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_Pr5160692531401422620nt_nat > fun_Pr3099820731930151597t_bool ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_int_nat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_Code_integer_nat > fun_Co8310303586393106575r_bool ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_complex_nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_Pr5160692531401422620nt_nat > fun_fu7672510408924296255t_bool ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_nat > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_Code_integer_nat > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_complex_nat > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: array_VEBT_VEBTi > fun_li7822782016457402668i_assn ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_VEBT_VEBT > fun_fu622286109622354600T_bool ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_VEBT_VEBT > fun_fu2956067580704323004T_bool ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_real > fun_fu6483983356097304844l_bool ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_real > fun_fu415297021002846548l_bool ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_nat > fun_fu6838017323210031556t_bool ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_nat > fun_fu3910677444521609704t_bool ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_int > fun_fu420584378438320396t_bool ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_int > fun_fu656925638523502036t_bool ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_Code_integer > fun_fu2342990897131189674r_bool ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_Code_integer > fun_fu4086510383517613953r_bool ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_complex > fun_fu7594865232509978120x_bool ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_complex > fun_fu3476074416393937614x_bool ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: set_Code_integer > fun_fu8681741693558125220r_bool ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_Code_integer > fun_fu3125884428166449269r_bool ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: code_integer > fun_fu3438678930028006669omplex ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: code_integer > fun_fu3438678930028006669omplex ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: set_nat > fun_fu6002750694894812635t_bool ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_nat > fun_fu1093554848263573315t_bool ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: set_nat > fun_fu1709340832601392443er_nat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_Co3312552843764780407at_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: set_Code_integer > fun_fu4694495798485950403r_bool ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: set_Code_integer > fun_fu5158082794216327598at_nat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: set_complex > fun_fu5159174019657214751x_bool ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: set_complex > fun_fu8973321760839499616omplex ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: set_VEBT_VEBT > fun_fu3138954370166276442omplex ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: set_complex > fun_fu2147946303232500784omplex ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: set_int > fun_fu2298032621795249330omplex ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: set_complex > fun_fu133147570795794627x_bool ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: set_complex > fun_fu2933621645842287103omplex ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_Co8905303033605938227omplex > fun_co1305099417915878579omplex ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: set_Code_integer > fun_fu4200121666219089987r_bool ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: set_Code_integer > fun_fu3955559650531061584omplex ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_co7673397065115134084omplex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: set_Code_integer > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_Co1900716531650368264omplex > fun_fu6624223949315877998omplex ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_Co3960896333048504454r_real > fun_fu6908834162160313196r_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_Co1900716531650368264omplex > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_Co3960896333048504454r_real > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_Code_integer_rat > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_Code_integer_nat > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: code_integer > fun_fu8135725022974991116r_real ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: code_integer > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: array_bool > fun_li6008471108080514341t_assn ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: array_nat > fun_li4928566814967809234t_assn ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: array_VEBT_VEBTi > fun_li7023630697154767927t_assn ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_VEBT_VEBT_nat > fun_fu5238640373165125763BT_nat ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: bool > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: bool > fun_Pr1572506220757514509t_bool ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: bool > fun_Nu28140169051909368851_bool ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: bool > fun_bool_bool ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_bo3288154511880838028t_bool ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_Pr803113274115125473t_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_int_bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_Co8822979462182099027r_bool > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_complex_bool > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_Pr1572506220757514509t_bool > fun_Pr1572506220757514509t_bool ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_Nu28140169051909368851_bool > fun_Nu28140169051909368851_bool ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_bool_bool > fun_bool_bool ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_int_bool ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_Pr1572506220757514509t_bool ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_Nu28140169051909368851_bool ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_real_bool ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_bool_bool ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_VE2989161857680785728t_real > fun_na2714374887907562620T_real ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: set_VEBT_VEBT > fun_fu830247258886439116t_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: set_real > fun_fu5104398059618711576t_real ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: set_int > fun_fu7141299348810748056t_real ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: set_nat > fun_fu7979797992566127108at_nat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_co3258106519541737638omplex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: set_complex > fun_fu689363623238371998omplex ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: set_nat > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: set_int > fun_fu688336635129761048at_int ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_real_real ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_int_int ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_nat_int ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_int_int ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: vEBT_VEBT > fun_fu2706034715333652099T_assn ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: real > fun_fu4701863041303578399l_assn ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: complex > fun_fu7489246418442965795x_assn ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: vEBT_VEBT > fun_fu2706034715333652099T_assn ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: real > fun_fu4701863041303578399l_assn ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: complex > fun_fu7489246418442965795x_assn ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_nat_fun_int_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_nat_fun_int_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_int_fun_nat_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_int_fun_nat_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: set_nat > fun_fu3446192182781468256BT_int ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_VE7331017413731640256at_int > fun_na3717937882823726332BT_int ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: set_VEBT_VEBT > fun_fu9054112635728933880at_int ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: set_nat > fun_fu6487107696295728876al_int ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: set_real > fun_fu1164691120062057232at_int ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: set_nat > fun_fu6449200239809977915er_int ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_Co822469698833053651at_int > fun_na3002214111362778963er_int ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: set_Code_integer > fun_fu2187553307443868334at_int ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: set_nat > fun_fu1666499901826068458ex_int ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: set_complex > fun_fu3438847080485518604at_int ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: set_int > fun_fu4313891142084863779t_bool ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: set_int > fun_fu3599772511451410272BT_int ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_VE294566951569886236nt_int > fun_in1463024724634152352BT_int ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: set_VEBT_VEBT > fun_fu3522331824740900180nt_int ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: set_int > fun_fu1082741970001819203t_bool ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: set_int > fun_fu3687342418701361539er_int ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_Co3009391273526075439nt_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: set_Code_integer > fun_fu5417119095707709635r_bool ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: set_Code_integer > fun_fu6891359585798312018nt_int ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: set_int > fun_fu4844541694308389507t_bool ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: set_complex > fun_fu8510821397121033091x_bool ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_nat_assn > fun_na2454419004871237530n_assn ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: fun_rat_rat ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Array__Time_Ofreeze_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Array__Time_Olen_001t__HOL__Obool,type,
    array_len_bool: array_bool > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Olen_001t__Nat__Onat,type,
    array_len_nat: array_nat > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Olen_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_len_VEBT_VEBTi: array_VEBT_VEBTi > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Onth_001t__HOL__Obool,type,
    array_nth_bool: ( array_bool * nat ) > heap_Time_Heap_bool ).

tff(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: ( array_nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    array_nth_option_nat: ( array_option_nat * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Array__Time_Oupd_001t__HOL__Obool,type,
    array_upd_bool: ( nat * bool * array_bool ) > heap_T6374562886178639292y_bool ).

tff(sy_c_Array__Time_Oupd_001t__Nat__Onat,type,
    array_upd_nat: ( nat * nat * array_nat ) > heap_T3836121109492952855ay_nat ).

tff(sy_c_Array__Time_Oupd_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun_Pr6001797050664311790t_bool ).

tff(sy_c_Assertions_Oentails,type,
    entails: fun_as7903617535776176121n_bool ).

tff(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: fun_li3352006551604695530i_assn > assn ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_bool_assn ).

tff(sy_c_Assertions_Osnga__assn_001t__HOL__Obool,type,
    snga_assn_bool: ( array_bool * list_bool ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: ( array_nat * list_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: ( array_option_nat * list_option_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > fun_as7903617535776176121n_bool ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: fun_co3258106519541737638omplex ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: fun_int_fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > fun_nat_rat ).

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > fun_num_num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun_Pr5837708615769641901m_bool ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bits__Integer_OBit__integer,type,
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tff(sy_c_Bits__Integer_Obin__last__integer,type,
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tff(sy_c_Bits__Integer_Obin__rest__integer,type,
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tff(sy_c_Code__Numeral_Odup,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > fun_real_complex ).

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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
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tff(sy_c_If_001t__Int__Oint,type,
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tff(sy_c_If_001t__Nat__Onat,type,
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tff(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_If_001t__Rat__Orat,type,
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tff(sy_c_If_001t__Real__Oreal,type,
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tff(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Int_Oint__ge__less__than,type,
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tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

tff(sy_c_Int_Onat,type,
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tff(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: fun_int_complex ).

tff(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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tff(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
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tff(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
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tff(sy_c_List_Ofoldr_001t__HOL__Obool_001t__Nat__Onat,type,
    foldr_bool_nat: ( fun_bool_fun_nat_nat * list_bool ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( fun_re7681423839302404990l_real * list_real ) > fun_real_real ).

tff(sy_c_List_Ofoldr_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
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tff(sy_c_List_Ofoldr_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
    map_int_real: ( fun_int_real * list_int ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( fun_nat_nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
    map_nat_real: ( fun_nat_real * list_nat ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
    map_real_real: ( fun_real_real * list_real ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_real_VEBT_VEBTi: ( fun_real_VEBT_VEBTi * list_real ) > list_VEBT_VEBTi ).

tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    map_real_VEBT_VEBT: ( fun_real_VEBT_VEBT * list_real ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    map_VEBT_VEBTi_real: ( fun_VEBT_VEBTi_real * list_VEBT_VEBTi ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE483055756984248624_VEBTi: ( fun_VE8449866691444886503_VEBTi * list_VEBT_VEBTi ) > list_VEBT_VEBTi ).

tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    map_VE7998069337340375161T_VEBT: ( fun_VE9120655849212850072T_VEBT * list_VEBT_VEBTi ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_VEBT_VEBT_real: ( fun_VEBT_VEBT_real * list_VEBT_VEBT ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE7029150624388687525_VEBTi: ( fun_VE712148701811721852_VEBTi * list_VEBT_VEBT ) > list_VEBT_VEBTi ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( fun_VE3165209603783583043T_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
    set_Code_integer2: list_Code_integer > set_Code_integer ).

tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
    set_bool2: list_bool > set_bool ).

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

tff(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
    set_VEBT_VEBTi2: list_VEBT_VEBTi > set_VEBT_VEBTi ).

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

tff(sy_c_List_Olist__update_001t__HOL__Obool,type,
    list_update_bool: ( list_bool * nat * bool ) > list_bool ).

tff(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: ( list_int * nat * int ) > list_int ).

tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: ( list_nat * nat * nat ) > list_nat ).

tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

tff(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: ( list_VEBT_VEBTi * nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: ( list_VEBT_VEBT * nat * vEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > fun_nat_Code_integer ).

tff(sy_c_List_Onth_001t__HOL__Obool,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > fun_nat_int ).

tff(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > fun_nat_nat ).

tff(sy_c_List_Onth_001t__Num__Onum,type,
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tff(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
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tff(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__HOL__Obool,type,
    product_real_bool: ( list_real * list_bool ) > list_P3020538015149218416l_bool ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oreplicate_001t__HOL__Obool,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: fun_complex_complex > fun_na1591161823130732966omplex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__HOL__Obool_J,type,
    size_size_list_bool: list_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
    size_size_list_num: list_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6086282163384603972on_nat: list_option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    size_s1877336372972134351al_nat: list_P6834414599653733731al_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    size_s3932428310213730859l_real: list_P8689742595348180415l_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    size_s6963394564282275252_VEBTi: list_P2340519324556330824_VEBTi > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s3289364478449617953T_VEBT: list_P877281246627933069T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    size_s5035110155006384947T_real: list_P2623026923184700063T_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    size_s3387956428969716412_VEBTi: list_P735349106241217576_VEBTi > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s7982070591426661849_VEBTi: list_VEBT_VEBTi > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_size_VEBT_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: fun_num_rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: fun_num_real ).

tff(sy_c_Num_Opow,type,
    pow: num > fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    none_P4506660739021792380nteger: option2651255830984564193nteger ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    none_P533106815845188193et_nat: option936205604648967762et_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    none_P4394680061957285238um_num: option2661157926820139483um_num ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__HOL__Obool_J,type,
    some_set_bool: set_bool > option_set_bool ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
    some_set_int: set_int > option_set_int ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Nat__Onat_J,type,
    bot_bot_option_nat: option_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HOL__Obool_J,type,
    bot_bot_set_bool: set_bool ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Assertions__Oassn,type,
    ord_less_assn: fun_as7903617535776176121n_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: ( option_int * option_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: ( option_num * option_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: ( option_real * option_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > fun_set_complex_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HOL__Obool_J,type,
    ord_less_set_bool: ( set_bool * set_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > fun_set_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > fun_set_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > fun_set_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
    ord_le4104064031414453916r_real: ( filter_real * filter_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_le1736525451366464988on_int: ( option_int * option_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: ( option_num * option_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_le2406147912482264968on_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le353528952715127954et_int: ( option_set_int * option_set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > fun_se2712834123562403997r_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    ord_le8840051076831633728t_bool: ( fun_nat_bool * fun_nat_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__HOL__Obool_J,type,
    ord_max_set_bool: ( set_bool * set_bool ) > set_bool ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__HOL__Obool_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Numeral____Type__Onum1_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Option__Ooption_It__Set__Oset_It__Real__Oreal_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    top_to4645266643341252675nteger: set_Code_integer ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    top_top_set_complex: set_complex ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
    top_top_set_bool: set_bool ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Code____Numeral__Ointeger_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Complex__Ocomplex_J_J,type,
    top_to6180147692022559204omplex: set_option_complex ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__HOL__Obool_J_J,type,
    top_to8850353440797054349n_bool: set_option_bool ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
    top_to6430115241214627170on_int: set_option_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    top_to8920198386146353926on_nat: set_option_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Numeral____Type__Onum1_J_J,type,
    top_to4428395536652758875l_num1: set_op4903093089046678027l_num1 ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Ounit_J_J,type,
    top_to2690860209552263555t_unit: set_op3165557761946182707t_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
    top_to853713521313446370n_real: set_option_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    top_to4366644338036079209nt_int: set_Pr958786334691620121nt_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > fun_nat_assn ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: fun_co3258106519541737638omplex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: ( heap_e7401611519738050253t_unit * set_nat ) > produc3658429121746597890et_nat ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > fun_in4211310926729683942nt_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: ( option_nat * option_nat ) > produc4953844613479565601on_nat ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: ( option4927543243414619207at_nat * option4927543243414619207at_nat ) > produc6121120109295599847at_nat ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: ( real * real ) > produc2422161461964618553l_real ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: ( real * vEBT_VEBT ) > produc3757001726724277373T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc3035007008238225530T_bool: ( vEBT_VEBT * bool ) > produc6448083994118892042T_bool ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: ( vEBT_VEBT * nat ) > produc9072475918466114483BT_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: ( vEBT_VEBT * real ) > produc5170161368751668367T_real ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc6084888613844515218_VEBTi: ( vEBT_VEBT * vEBT_VEBTi ) > produc3625547720036274456_VEBTi ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc2004789812759622991on_nat: ( fun_nat_fun_nat_bool * produc4953844613479565601on_nat ) > produc3700678068921409119on_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2818902965508115497at_nat: ( fun_nat_fun_nat_nat * produc7248412053542808358at_nat ) > produc9217583074051984497at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc6699827698519632094on_nat: ( fun_nat_fun_nat_nat * produc4953844613479565601on_nat ) > produc7004362771917097900on_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc5961576194287291153at_nat: ( fun_Pr248683899324676269t_bool * produc6121120109295599847at_nat ) > produc533602882674032673at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7814092408141220007at_nat: ( fun_Pr2477239776880934807at_nat * produc6121120109295599847at_nat ) > produc1260151622200656695at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
    produc8560705615207772305t_bool: fun_int_fun_int_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( fun_na4138050610385705875at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

tff(sy_c_Rat_OFrct,type,
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tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__HOL__Obool,type,
    refine4616567020171210338s_bool: ( heap_Time_Heap_bool * heap_Time_Heap_bool ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__HOL__Obool_J,type,
    refine3902108050975373032t_bool: ( heap_T5178755288483056679t_bool * heap_T5178755288483056679t_bool ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    refine1935026298455697829on_nat: ( heap_T5317711798761887292on_nat * heap_T5317711798761887292on_nat ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_Refine__Imp__Hol_Orefines_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Assertions__Oassn,type,
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    time_time_nat: ( heap_Time_Heap_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: ( heap_T2636463487746394924on_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: fun_VE6376819126223186655i_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__HOL__Obool,type,
    vEBT_V6044891547498605301i_bool: ( nat * heap_Time_Heap_bool ) > heap_T5178755288483056679t_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: ( nat * heap_Time_Heap_nat ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: ( nat * heap_T2636463487746394924on_nat ) > heap_T5317711798761887292on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: ( nat * heap_T8145700208782473153_VEBTi ) > heap_T4980287057938770641_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( bool * bool ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option4927543243414619207at_nat * nat * array_VEBT_VEBTi ) > fun_VE8449866691444886503_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__HOL__Obool_J,type,
    vEBT_c4579152981172362421p_bool: ( fun_op3392518339485081117p_bool * fun_bo7602104967519523767p_bool * vEBT_VEBTi ) > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( fun_op2425996437410981688ap_nat * fun_bo5340758216624605938ap_nat * vEBT_VEBTi ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( fun_op5783775701855621896on_nat * fun_bo8430231047727935170on_nat * vEBT_VEBTi ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( fun_op5127434927081926045_VEBTi * fun_bo851793448214054487_VEBTi * vEBT_VEBTi ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( fun_op7256438278201533260Ti_nat * fun_bo5748384255075588070ol_nat ) > fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr9085508990737135641i_bool ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun_VE6376819126223186655i_bool ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun_VE6376819126223186655i_bool ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: bool > fun_bool_VEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__HOL__Obool_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6372884366951783724_VEBTi: ( set_nat * fun_bo7535889243153246212i_assn * list_bool ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2992182275864018813T_VEBT: ( set_nat * fun_bo5394993412183903963T_assn * list_bool * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__HOL__Obool,type,
    vEBT_L3606510757799916617l_bool: ( set_nat * fun_re2206215847901373695l_assn * list_real * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: ( set_nat * fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L5184575500739366650l_real: ( set_nat * fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: ( set_nat * fun_re560840401387337987i_assn * list_real ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: ( set_nat * fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    vEBT_L6176004506706552451T_bool: ( set_nat * fun_VE1323945509527881313l_assn * list_VEBT_VEBT * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: ( set_nat * fun_VE4931415397359990460t_assn * list_VEBT_VEBT * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: ( set_nat * fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: ( set_nat * fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Real__Oreal,type,
    vEBT_L5107238702033023424l_real: ( fun_bo8885878543415060347l_assn * list_bool * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2136499394512906784T_VEBT: ( fun_bo5394993412183903963T_assn * list_bool * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L74593716426352029nt_int: ( fun_int_fun_int_assn * list_int * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L77084186935402305nt_nat: ( fun_int_fun_nat_assn * list_int * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( fun_in6446958447386763770l_assn * list_int * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1664421287176695555T_VEBT: ( fun_in7755481285845205148T_assn * list_int * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( fun_na448437756743965086l_assn * list_nat * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8158188754432654943T_VEBT: ( fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__HOL__Obool,type,
    vEBT_L870199270341830246l_bool: ( fun_re2206215847901373695l_assn * list_real * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Int__Oint,type,
    vEBT_L1443519841834266653al_int: ( fun_re4990660043736809722t_assn * list_real * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( fun_re560840401387337987i_assn * list_real ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L8937798142398754470i_real: ( fun_VE1885522831824361987l_assn * list_VEBT_VEBTi * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    vEBT_L5320321625355440422T_bool: ( fun_VE1323945509527881313l_assn * list_VEBT_VEBT ) > fun_list_bool_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( fun_VE123837653852366616t_assn * list_VEBT_VEBT * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( fun_VE4931415397359990460t_assn * list_VEBT_VEBT ) > fun_list_nat_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( fun_VE6334927249922962060t_assn * list_VEBT_VEBT ) > fun_li338670087962420949t_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > fun_nat_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option_nat ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr3099820731930151597t_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: fun_Pr5837708615769641901m_bool > fun_Pr6287530515171585797m_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr817965127378462947t_bool * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( fun_Pr9085508990737135641i_bool * produc3625547720036274456_VEBTi ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: fun_VE6376819126223186655i_bool > fun_VEBT_VEBTi_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE5877677516038742509T_bool > fun_VEBT_VEBT_bool ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    aa_assn_assn: ( fun_assn_assn * assn ) > assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__HOL__Obool,type,
    aa_assn_bool: ( fun_assn_bool * assn ) > bool ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Assertions__Oassn_Mt__HOL__Obool_J,type,
    aa_ass5323576183532441186n_bool: ( fun_as7903617535776176121n_bool * assn ) > fun_assn_bool ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__HOL__Obool_Mt__Assertions__Oassn_J,type,
    aa_ass2376637134808068204l_assn: ( fun_as6968651184343528707l_assn * assn ) > fun_bool_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__Assertions__Oassn_J,type,
    aa_ass1897699608592665680i_assn: ( fun_as5323700489645809679i_assn * assn ) > fun_li3352006551604695530i_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    aa_assn_fun_nat_assn: ( fun_as8240436605320707994t_assn * assn ) > fun_nat_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J,type,
    aa_ass4112278883933122603t_assn: ( fun_as2043416024874063466t_assn * assn ) > fun_option_nat_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J,type,
    aa_ass4389913640333755968i_assn: ( fun_as315122898794150911i_assn * assn ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn,type,
    aa_Code_integer_assn: ( fun_Co4490034186337852034r_assn * code_integer ) > assn ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    aa_Cod1783974652533396728nteger: ( fun_Co8499238673658658103nteger * code_integer ) > code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Complex__Ocomplex,type,
    aa_Cod5621350732091595593omplex: ( fun_Co1900716531650368264omplex * code_integer ) > complex ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
    aa_Code_integer_bool: ( fun_Co8822979462182099027r_bool * code_integer ) > bool ).

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tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun6018615163312975129t_assn: ( fun_fu1092876921774764064t_assn * fun_VE4931415397359990460t_assn ) > fun_VE7969100967704966681t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun1384843155787029862at_int: ( fun_fu5075381843755512741at_int * fun_VE3763956218935105199t_bool ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun5562694175296226570at_nat: ( fun_fu7565464988687239497at_nat * fun_VE3763956218935105199t_bool ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    aa_fun4018725027000996448T_bool: ( fun_fu2721280555464079775T_bool * fun_VE3763956218935105199t_bool ) > fun_na2351039247863542899T_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    aa_fun6141810409722670662BT_int: ( fun_fu697195148766750301BT_int * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    aa_fun1096289392377091562BT_nat: ( fun_fu3187278293698477057BT_nat * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    aa_fun5431641998072558748t_bool: ( fun_fu6002750694894812635t_bool * fun_VE3763956218935105199t_bool ) > fun_VE3763956218935105199t_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun6066863168309489065at_int: ( fun_fu9054112635728933880at_int * fun_VE7331017413731640256at_int ) > fun_fu5075381843755512741at_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    aa_fun6474440919679335769BT_int: ( fun_fu3446192182781468256BT_int * fun_VE7331017413731640256at_int ) > fun_fu697195148766750301BT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun8318684123515291377at_nat: ( fun_fu4406026827936152384at_nat * fun_VE597728521808591204at_nat ) > fun_fu7565464988687239497at_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    aa_fun7476968357888788569BT_nat: ( fun_fu6772184894847113056BT_nat * fun_VE597728521808591204at_nat ) > fun_fu3187278293698477057BT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun6147343916763992189t_real: ( fun_fu830247258886439116t_real * fun_VE2989161857680785728t_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun7129419364150962969t_assn: ( fun_fu8831737957336725536t_assn * fun_VE6334927249922962060t_assn ) > fun_VE2615472500133170025t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J,type,
    aa_fun1240582285364316732t_assn: ( fun_fu8734043002210220995t_assn * fun_VE6334927249922962060t_assn ) > fun_VE6334927249922962060t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun8635824850342594777i_assn: ( fun_fu8850137702660867424i_assn * fun_VE905575822219544865i_assn ) > fun_VE2880048358281379966i_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J,type,
    aa_fun4659777589554175228i_assn: ( fun_fu8979060164380311043i_assn * fun_VE905575822219544865i_assn ) > fun_VE905575822219544865i_assn ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_bool > real ).

tff(sy_c_fEx_001t__Real__Oreal,type,
    fEx_real: fun_fu7511176317168184086l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: bool > fun_bool_bool ).

tff(sy_c_fdisj,type,
    fdisj: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
    fequal_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1378794544173401837x_bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Heap__Oarray_It__HOL__Obool_J,type,
    fequal_array_bool: ( array_bool * array_bool ) > bool ).

tff(sy_c_fequal_001t__Heap__Oarray_It__Nat__Onat_J,type,
    fequal_array_nat: ( array_nat * array_nat ) > bool ).

tff(sy_c_fequal_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    fequal8089995073349378887_VEBTi: ( array_VEBT_VEBTi * array_VEBT_VEBTi ) > bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    fequal1389750275112723248_VEBTi: list_VEBT_VEBTi > fun_li5928143645361859307i_bool ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Nat__Onat_J,type,
    fequal_option_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fequal3124746952608472732at_nat: fun_op5682630191948829869t_bool ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    fequal6010348034681699726nt_int: product_prod_int_int > fun_Pr803113274115125473t_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fequal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    fequal_VEBT_VEBTi: vEBT_VEBTi > fun_VEBT_VEBTi_bool ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: vEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_fimplies,type,
    fimplies: bool > fun_bool_bool ).

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > fun_se2712834123562403997r_bool ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > fun_set_complex_bool ).

tff(sy_c_member_001t__HOL__Obool,type,
    member_bool: ( bool * set_bool ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: int > fun_set_int_bool ).

tff(sy_c_member_001t__List__Olist_It__HOL__Obool_J,type,
    member_list_bool: ( list_bool * set_list_bool ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: ( list_real * set_list_real ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    member8117914210271334748_VEBTi: ( list_VEBT_VEBTi * set_list_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_bool ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: num > fun_set_num_bool ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: ( produc8923325533196201883nteger * set_Pr4811707699266497531nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: ( produc3658429121746597890et_nat * set_Pr3948176798113811640et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > fun_set_rat_bool ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: real > fun_set_real_bool ).

tff(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: ( set_int * set_set_int ) > $o ).

tff(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: ( vEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > fun_se448528567104302406T_bool ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_aktnode____,type,
    aktnode: vEBT_VEBT ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_minew____,type,
    minew: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_newnode____,type,
    newnode: vEBT_VEBT ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_tree__is______,type,
    tree_is: list_VEBT_VEBTi ).

tff(sy_v_va____,type,
    va: nat ).

tff(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

tff(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_xaa______,type,
    xaa: vEBT_VEBT ).

tff(sy_v_xb______,type,
    xb: vEBT_VEBTi ).

tff(sy_v_xba______,type,
    xba: option_nat ).

tff(sy_v_xc,type,
    xc: vEBT_VEBTi ).

tff(sy_v_xnew____,type,
    xnew: nat ).

% Relevant facts (11167)
tff(fact_0_groupy,axiom,
    ! [A: assn,B: assn,C: assn,D: assn,X: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),aa_assn_assn(times_times_assn(C),D))),X))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C)),D)),X)) ) ).

% groupy
tff(fact_1_midextr,axiom,
    ! [P: assn,Q: assn,Q2: assn,R: assn,X: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),Q2)),R)),X))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),R)),Q)),Q2)),X)) ) ).

% midextr
tff(fact_2_swappa,axiom,
    ! [B: assn,A: assn,C: assn,X: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(B),A)),C)),X))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C)),X)) ) ).

% swappa
tff(fact_3_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_4_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_5_bit__split__inv,axiom,
    ! [X2: nat,D2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),D2)),D2) = X2 ).

% bit_split_inv
tff(fact_6_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is)),aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is)))) ).

% assnle
tff(fact_7_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_8_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_9_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2)),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_10_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,X2)),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_11_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X2)),I))
      <=> pp(X2) ) ) ).

% nth_list_update_eq
tff(fact_12_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_13_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,X2: vEBT_VEBTi] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J,X2),I) = nth_VEBT_VEBTi(L,I) ) ) ).

% nth_update_invalid
tff(fact_14_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J,X2)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I) ) ) ).

% nth_update_invalid
tff(fact_15_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,X2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( aa_nat_real(nth_real(list_update_real(L,J,X2)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_16_nth__update__invalid,axiom,
    ! [I: nat,L: list_bool,J: nat,X2: bool] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(L,J,X2)),I))
      <=> pp(aa_nat_bool(nth_bool(L),I)) ) ) ).

% nth_update_invalid
tff(fact_17_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,X2)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_18_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_19_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_20_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))),B2)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_21_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W))),B2)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_22_delt__out__of__range,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% delt_out_of_range
tff(fact_23_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] : list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,I)) = Xs ).

% list_update_id
tff(fact_24_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)) = Xs ).

% list_update_id
tff(fact_25_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_26_list__update__id,axiom,
    ! [Xs: list_bool,I: nat] : list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),I)) = Xs ).

% list_update_id
tff(fact_27_list__update__id,axiom,
    ! [Xs: list_real,I: nat] : list_update_real(Xs,I,aa_nat_real(nth_real(Xs),I)) = Xs ).

% list_update_id
tff(fact_28_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( I != J )
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2),J) = nth_VEBT_VEBTi(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_29_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( I != J )
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2)),J) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_30_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X2: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_31_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_bool,X2: bool] :
      ( ( I != J )
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X2)),J))
      <=> pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ).

% nth_list_update_neq
tff(fact_32_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_real,X2: real] :
      ( ( I != J )
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,X2)),J) = aa_nat_real(nth_real(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_33_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_34_length__list__update,axiom,
    ! [Xs: list_real,I: nat,X2: real] : size_size_list_real(list_update_real(Xs,I,X2)) = size_size_list_real(Xs) ).

% length_list_update
tff(fact_35_length__list__update,axiom,
    ! [Xs: list_bool,I: nat,X2: bool] : size_size_list_bool(list_update_bool(Xs,I,X2)) = size_size_list_bool(Xs) ).

% length_list_update
tff(fact_36_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] : size_size_list_nat(list_update_nat(Xs,I,X2)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_37_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_38_not__None__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( X2 != none_P5556105721700978146at_nat )
    <=> ? [Y: product_prod_nat_nat] : X2 = some_P7363390416028606310at_nat(Y) ) ).

% not_None_eq
tff(fact_39_not__None__eq,axiom,
    ! [X2: option_nat] :
      ( ( X2 != none_nat )
    <=> ? [Y: nat] : X2 = some_nat(Y) ) ).

% not_None_eq
tff(fact_40_not__Some__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ! [Y: product_prod_nat_nat] : X2 != some_P7363390416028606310at_nat(Y)
    <=> ( X2 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_41_not__Some__eq,axiom,
    ! [X2: option_nat] :
      ( ! [Y: nat] : X2 != some_nat(Y)
    <=> ( X2 = none_nat ) ) ).

% not_Some_eq
tff(fact_42_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_43_max__in__set__def,axiom,
    ! [Xs: set_nat,X2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(Xs),X2))
    <=> ( pp(aa_set_nat_bool(member_nat(X2),Xs))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X2)) ) ) ) ).

% max_in_set_def
tff(fact_44_min__in__set__def,axiom,
    ! [Xs: set_nat,X2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(Xs),X2))
    <=> ( pp(aa_set_nat_bool(member_nat(X2),Xs))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),X3)) ) ) ) ).

% min_in_set_def
tff(fact_45_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_46_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_47_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,M) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_48_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_49_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_50_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_51_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_52_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_53_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( some_nat(X22) = some_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_54_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi,Y3: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2),I,Y3) = list_u6098035379799741383_VEBTi(Xs,I,Y3) ).

% list_update_overwrite
tff(fact_55_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT,Y3: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),I,Y3) = list_u1324408373059187874T_VEBT(Xs,I,Y3) ).

% list_update_overwrite
tff(fact_56_list__update__overwrite,axiom,
    ! [Xs: list_nat,I: nat,X2: nat,Y3: nat] : list_update_nat(list_update_nat(Xs,I,X2),I,Y3) = list_update_nat(Xs,I,Y3) ).

% list_update_overwrite
tff(fact_57_list__update__overwrite,axiom,
    ! [Xs: list_bool,I: nat,X2: bool,Y3: bool] : list_update_bool(list_update_bool(Xs,I,X2),I,Y3) = list_update_bool(Xs,I,Y3) ).

% list_update_overwrite
tff(fact_58_list__update__overwrite,axiom,
    ! [Xs: list_real,I: nat,X2: real,Y3: real] : list_update_real(list_update_real(Xs,I,X2),I,Y3) = list_update_real(Xs,I,Y3) ).

% list_update_overwrite
tff(fact_59_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_60_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_61_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_62_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_63_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_64_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_65_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_66_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_67_num__double,axiom,
    ! [N: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_68_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_69_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_70_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_71_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_72_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_73_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_74_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_75_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_76_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_77_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_78_del__single__cont,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( X2 = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ) ) ) ).

% del_single_cont
tff(fact_79_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_80_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_81_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_82_not__Some__eq2,axiom,
    ! [V: option2651255830984564193nteger] :
      ( ! [X3: code_integer,Y: code_integer] : V != some_P6772290148444788224nteger(produc1086072967326762835nteger(X3,Y))
    <=> ( V = none_P4506660739021792380nteger ) ) ).

% not_Some_eq2
tff(fact_83_not__Some__eq2,axiom,
    ! [V: option936205604648967762et_nat] :
      ( ! [X3: heap_e7401611519738050253t_unit,Y: set_nat] : V != some_P624177172695371229et_nat(produc7507926704131184380et_nat(X3,Y))
    <=> ( V = none_P533106815845188193et_nat ) ) ).

% not_Some_eq2
tff(fact_84_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ! [X3: num,Y: num] : V != some_P6201964756284913402um_num(product_Pair_num_num(X3,Y))
    <=> ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
tff(fact_85_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X3: int,Y: int] : V != some_P4184893108420464158nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_86_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X3: nat,Y: nat] : V != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_87_lesseq__shift,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,some_nat(X2)),some_nat(Y3))) ) ).

% lesseq_shift
tff(fact_88_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
     => ( list_u1324408373059187874T_VEBT(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_89_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_real(Xs)),I))
     => ( list_update_real(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_90_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I))
     => ( list_update_bool(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_91_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
     => ( list_update_nat(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_92_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I))
     => ( list_u6098035379799741383_VEBTi(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_93_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))),B2)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_94_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W))),B2)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_95_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_96_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_97_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N3)),aa_nat_set_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_98_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_99_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_100_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_101_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_102_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N3))),aa_nat_set_int(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_103_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N3))),aa_nat_rat(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_104_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N3))),aa_nat_num(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_105_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N3))),aa_nat_nat(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_106_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N3))),aa_nat_int(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_107_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_108_mem__Collect__eq,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),collect_VEBT_VEBT(P)))
    <=> pp(aa_VEBT_VEBT_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_109_mem__Collect__eq,axiom,
    ! [A2: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A2),collect_real(P)))
    <=> pp(aa_real_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_110_mem__Collect__eq,axiom,
    ! [A2: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A2),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_111_mem__Collect__eq,axiom,
    ! [A2: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A2),collect_int(P)))
    <=> pp(aa_int_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_112_mem__Collect__eq,axiom,
    ! [A2: complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(A2),collect_complex(P)))
    <=> pp(aa_complex_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_113_mem__Collect__eq,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr803113274115125473t_bool] :
      ( member5262025264175285858nt_int(A2,collec213857154873943460nt_int(P))
    <=> pp(aa_Pro4501881432599481050t_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_114_Collect__mem__eq,axiom,
    ! [A: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_a(A)) = A ).

% Collect_mem_eq
tff(fact_115_Collect__mem__eq,axiom,
    ! [A: set_real] : collect_real(aTP_Lamm_aa(A)) = A ).

% Collect_mem_eq
tff(fact_116_Collect__mem__eq,axiom,
    ! [A: set_nat] : collect_nat(aTP_Lamm_ab(A)) = A ).

% Collect_mem_eq
tff(fact_117_Collect__mem__eq,axiom,
    ! [A: set_int] : collect_int(aTP_Lamm_ac(A)) = A ).

% Collect_mem_eq
tff(fact_118_Collect__mem__eq,axiom,
    ! [A: set_complex] : collect_complex(aTP_Lamm_ad(A)) = A ).

% Collect_mem_eq
tff(fact_119_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ae(A)) = A ).

% Collect_mem_eq
tff(fact_120_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
        <=> pp(aa_nat_bool(Q,X4)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_121_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(Q,X4)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_122_Collect__cong,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X4: complex] :
          ( pp(aa_complex_bool(P,X4))
        <=> pp(aa_complex_bool(Q,X4)) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_123_Collect__cong,axiom,
    ! [P: fun_Pr803113274115125473t_bool,Q: fun_Pr803113274115125473t_bool] :
      ( ! [X4: product_prod_int_int] :
          ( pp(aa_Pro4501881432599481050t_bool(P,X4))
        <=> pp(aa_Pro4501881432599481050t_bool(Q,X4)) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_124_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_125_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_126_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_127_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_128_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_129_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_130_le__some__optE,axiom,
    ! [M: set_int,X2: option_set_int] :
      ( ord_le353528952715127954et_int(some_set_int(M),X2)
     => ~ ! [M2: set_int] :
            ( ( X2 = some_set_int(M2) )
           => ~ pp(aa_set_int_bool(ord_less_eq_set_int(M),M2)) ) ) ).

% le_some_optE
tff(fact_131_le__some__optE,axiom,
    ! [M: rat,X2: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),X2)
     => ~ ! [M2: rat] :
            ( ( X2 = some_rat(M2) )
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M),M2)) ) ) ).

% le_some_optE
tff(fact_132_le__some__optE,axiom,
    ! [M: num,X2: option_num] :
      ( ord_le6622620407824499402on_num(some_num(M),X2)
     => ~ ! [M2: num] :
            ( ( X2 = some_num(M2) )
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),M2)) ) ) ).

% le_some_optE
tff(fact_133_le__some__optE,axiom,
    ! [M: nat,X2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,some_nat(M)),X2))
     => ~ ! [M2: nat] :
            ( ( X2 = some_nat(M2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),M2)) ) ) ).

% le_some_optE
tff(fact_134_le__some__optE,axiom,
    ! [M: int,X2: option_int] :
      ( ord_le1736525451366464988on_int(some_int(M),X2)
     => ~ ! [M2: int] :
            ( ( X2 = some_int(M2) )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),M2)) ) ) ).

% le_some_optE
tff(fact_135_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_136_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_137_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_138_ord__eq__le__eq__trans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int,D2: set_int] :
      ( ( A2 = B2 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => ( ( C2 = D2 )
         => pp(aa_set_int_bool(ord_less_eq_set_int(A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_139_ord__eq__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( A2 = B2 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ( C2 = D2 )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_140_ord__eq__le__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( A2 = B2 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ( C2 = D2 )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_141_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( A2 = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => ( ( C2 = D2 )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_142_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( A2 = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => ( ( C2 = D2 )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_143_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_144_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),B2)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
            & ! [Y5: nat] :
                ( pp(aa_nat_bool(P,Y5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y5),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_145_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_146_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_147_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_148_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y4: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y4))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y4),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z2)) ) )
         => ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N3),aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_149_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N3))
             => ( pp(aa_nat_bool(P,N3))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_150_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M3)),N3))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N3)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_151_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_152_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_153_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_154_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M4))
     => ? [M5: nat] : M4 = aa_nat_nat(suc,M5) ) ).

% Suc_le_D
tff(fact_155_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_156_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_157_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_158_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_159_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_160_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_161_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_162_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_163_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_164_exists__leI,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ( ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),N))
           => ~ pp(aa_nat_bool(P,N4)) )
       => pp(aa_nat_bool(P,N)) )
     => ? [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N))
          & pp(aa_nat_bool(P,N5)) ) ) ).

% exists_leI
tff(fact_165_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q3) ).

% div_mult2_eq
tff(fact_166_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).

% div_le_dividend
tff(fact_167_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).

% div_le_mono
tff(fact_168_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q3))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_169_div__mult2__numeral__eq,axiom,
    ! [A2: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_170_div__mult2__numeral__eq,axiom,
    ! [A2: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_171_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_172_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_173_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_174_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_175_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_176_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_177_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N3)))
                 => pp(aa_nat_bool(P,N3)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_178_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J))
               => ( pp(aa_nat_bool(P,N3))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_179_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_180_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_181_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(suc,A2))) )
    <=> ( B2 = aa_nat_nat(suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_182_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(suc,A2))) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_183_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_184_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_185_bex2I,axiom,
    ! [A2: code_integer,B2: code_integer,S: set_Pr4811707699266497531nteger,P: fun_Co8310303586393106575r_bool] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(A2,B2),S)
     => ( ( member157494554546826820nteger(produc1086072967326762835nteger(A2,B2),S)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A2),B2)) )
       => ? [A3: code_integer,B3: code_integer] :
            ( member157494554546826820nteger(produc1086072967326762835nteger(A3,B3),S)
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_186_bex2I,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,S: set_Pr3948176798113811640et_nat,P: fun_he7710493197641524256t_bool] :
      ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A2,B2),S)
     => ( ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A2,B2),S)
         => pp(aa_set_nat_bool(aa_hea2782697410307197849t_bool(P,A2),B2)) )
       => ? [A3: heap_e7401611519738050253t_unit,B3: set_nat] :
            ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A3,B3),S)
            & pp(aa_set_nat_bool(aa_hea2782697410307197849t_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_187_bex2I,axiom,
    ! [A2: num,B2: num,S: set_Pr8218934625190621173um_num,P: fun_num_fun_num_bool] :
      ( member7279096912039735102um_num(product_Pair_num_num(A2,B2),S)
     => ( ( member7279096912039735102um_num(product_Pair_num_num(A2,B2),S)
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) )
       => ? [A3: num,B3: num] :
            ( member7279096912039735102um_num(product_Pair_num_num(A3,B3),S)
            & pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_188_bex2I,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_bool] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),S)
     => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),S)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) )
       => ? [A3: nat,B3: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3),S)
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_189_bex2I,axiom,
    ! [A2: int,B2: int,S: set_Pr958786334691620121nt_int,P: fun_int_fun_int_bool] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),S)
     => ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),S)
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) )
       => ? [A3: int,B3: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3),S)
            & pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_190_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_191_Suc__inject,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y3) )
     => ( X2 = Y3 ) ) ).

% Suc_inject
tff(fact_192_linorder__neqE__nat,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ) ).

% linorder_neqE_nat
tff(fact_193_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ~ pp(aa_nat_bool(P,N3))
         => ? [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
              & ~ pp(aa_nat_bool(P,M3)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_194_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N3)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_195_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_196_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
     => ( S2 != T ) ) ).

% less_not_refl3
tff(fact_197_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_198_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_199_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_200_size__neq__size__imp__neq,axiom,
    ! [X2: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X2) != size_s6755466524823107622T_VEBT(Y3) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_201_size__neq__size__imp__neq,axiom,
    ! [X2: list_real,Y3: list_real] :
      ( ( size_size_list_real(X2) != size_size_list_real(Y3) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_202_size__neq__size__imp__neq,axiom,
    ! [X2: list_bool,Y3: list_bool] :
      ( ( size_size_list_bool(X2) != size_size_list_bool(Y3) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_203_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( size_size_list_nat(X2) != size_size_list_nat(Y3) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_204_size__neq__size__imp__neq,axiom,
    ! [X2: num,Y3: num] :
      ( ( aa_num_nat(size_size_num,X2) != aa_num_nat(size_size_num,Y3) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_205_size__neq__size__imp__neq,axiom,
    ! [X2: list_VEBT_VEBTi,Y3: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(X2) != size_s7982070591426661849_VEBTi(Y3) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
tff(fact_206_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_207_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) != size_size_list_real(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_208_neq__if__length__neq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) != size_size_list_bool(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_209_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_210_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) != size_s7982070591426661849_VEBTi(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_211_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).

% Ex_list_of_length
tff(fact_212_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_real] : size_size_list_real(Xs2) = N ).

% Ex_list_of_length
tff(fact_213_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_214_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_215_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBTi] : size_s7982070591426661849_VEBTi(Xs2) = N ).

% Ex_list_of_length
tff(fact_216_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_VEBT_VEBTi,X2: vEBT_VEBTi,X5: vEBT_VEBTi] :
      ( ( I != I3 )
     => ( list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2),I3,X5) = list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I3,X5),I,X2) ) ) ).

% list_update_swap
tff(fact_217_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I != I3 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),I3,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,X5),I,X2) ) ) ).

% list_update_swap
tff(fact_218_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_nat,X2: nat,X5: nat] :
      ( ( I != I3 )
     => ( list_update_nat(list_update_nat(Xs,I,X2),I3,X5) = list_update_nat(list_update_nat(Xs,I3,X5),I,X2) ) ) ).

% list_update_swap
tff(fact_219_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_bool,X2: bool,X5: bool] :
      ( ( I != I3 )
     => ( list_update_bool(list_update_bool(Xs,I,X2),I3,X5) = list_update_bool(list_update_bool(Xs,I3,X5),I,X2) ) ) ).

% list_update_swap
tff(fact_220_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_real,X2: real,X5: real] :
      ( ( I != I3 )
     => ( list_update_real(list_update_real(Xs,I,X2),I3,X5) = list_update_real(list_update_real(Xs,I3,X5),I,X2) ) ) ).

% list_update_swap
tff(fact_221_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_222_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => pp(aa_nat_bool(P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
               => pp(aa_nat_bool(P,I2)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_223_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_224_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_225_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_226_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_227_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M6: nat] :
          ( ( M = aa_nat_nat(suc,M6) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M6)) ) ) ).

% Suc_less_eq2
tff(fact_228_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,I4)) ) ) ) ).

% Nat.All_less_Suc
tff(fact_229_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_230_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_231_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_232_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_233_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_234_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_235_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_236_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_237_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_238_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_239_length__induct,axiom,
    ! [P: fun_list_real_bool,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2)))
             => pp(aa_list_real_bool(P,Ys2)) )
         => pp(aa_list_real_bool(P,Xs2)) )
     => pp(aa_list_real_bool(P,Xs)) ) ).

% length_induct
tff(fact_240_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys2: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys2)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P,Ys2)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_241_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_242_length__induct,axiom,
    ! [P: fun_li5928143645361859307i_bool,Xs: list_VEBT_VEBTi] :
      ( ! [Xs2: list_VEBT_VEBTi] :
          ( ! [Ys2: list_VEBT_VEBTi] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s7982070591426661849_VEBTi(Ys2)),size_s7982070591426661849_VEBTi(Xs2)))
             => pp(aa_lis8521252006704567508i_bool(P,Ys2)) )
         => pp(aa_lis8521252006704567508i_bool(P,Xs2)) )
     => pp(aa_lis8521252006704567508i_bool(P,Xs)) ) ).

% length_induct
tff(fact_243_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: fun_op5682630191948829869t_bool,Y3: option4927543243414619207at_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X2),Y3)) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X2),Y3)) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X2 = some_P7363390416028606310at_nat(A3) )
             => ( ( Y3 = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X2),Y3)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X2),Y3)) ) ) ) ).

% combine_options_cases
tff(fact_244_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: fun_op9100744243108201484t_bool,Y3: option_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X2),Y3)) )
     => ( ( ( Y3 = none_nat )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X2),Y3)) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X2 = some_P7363390416028606310at_nat(A3) )
             => ( ( Y3 = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X2),Y3)) ) )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X2),Y3)) ) ) ) ).

% combine_options_cases
tff(fact_245_combine__options__cases,axiom,
    ! [X2: option_nat,P: fun_op5698422665285572208t_bool,Y3: option4927543243414619207at_nat] :
      ( ( ( X2 = none_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X2),Y3)) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X2),Y3)) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X2 = some_nat(A3) )
             => ( ( Y3 = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X2),Y3)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X2),Y3)) ) ) ) ).

% combine_options_cases
tff(fact_246_combine__options__cases,axiom,
    ! [X2: option_nat,P: fun_op7792301606653096457t_bool,Y3: option_nat] :
      ( ( ( X2 = none_nat )
       => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X2),Y3)) )
     => ( ( ( Y3 = none_nat )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X2),Y3)) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X2 = some_nat(A3) )
             => ( ( Y3 = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X2),Y3)) ) )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X2),Y3)) ) ) ) ).

% combine_options_cases
tff(fact_247_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_1: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_1))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        & ! [X3: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X3))) ) ) ).

% split_option_all
tff(fact_248_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_1: option_nat] : pp(aa_option_nat_bool(P,X_1))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_all
tff(fact_249_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_1: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_1))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        | ? [X3: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X3))) ) ) ).

% split_option_ex
tff(fact_250_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_1: option_nat] : pp(aa_option_nat_bool(P,X_1))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_ex
tff(fact_251_option_Oexhaust,axiom,
    ! [Y3: option4927543243414619207at_nat] :
      ( ( Y3 != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y3 != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_252_option_Oexhaust,axiom,
    ! [Y3: option_nat] :
      ( ( Y3 != none_nat )
     => ~ ! [X23: nat] : Y3 != some_nat(X23) ) ).

% option.exhaust
tff(fact_253_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_254_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_255_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_256_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_257_mult__numeral__1__right,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(one)) = A2 ).

% mult_numeral_1_right
tff(fact_258_mult__numeral__1__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,one)) = A2 ).

% mult_numeral_1_right
tff(fact_259_mult__numeral__1__right,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,one)) = A2 ).

% mult_numeral_1_right
tff(fact_260_mult__numeral__1__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_num_nat(numeral_numeral_nat,one)) = A2 ).

% mult_numeral_1_right
tff(fact_261_mult__numeral__1__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_num_int(numeral_numeral_int,one)) = A2 ).

% mult_numeral_1_right
tff(fact_262_mult__numeral__1,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(one)),A2) = A2 ).

% mult_numeral_1
tff(fact_263_mult__numeral__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_264_mult__numeral__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_265_mult__numeral__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_266_mult__numeral__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_267_divide__numeral__1,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),numera6690914467698888265omplex(one)) = A2 ).

% divide_numeral_1
tff(fact_268_divide__numeral__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,one)) = A2 ).

% divide_numeral_1
tff(fact_269_divide__numeral__1,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),aa_num_rat(numeral_numeral_rat,one)) = A2 ).

% divide_numeral_1
tff(fact_270_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_271_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_272_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N3)),aa_nat_num(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_273_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_274_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N3)),aa_nat_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_275_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_276_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_277_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N3)),aa_nat_num(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_278_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_279_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N3)),aa_nat_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_280_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE3763956218935105199t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: vEBT_VEBT] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(P,Li),I2)) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L2) = N )
           => ~ ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L2),I5)),I5)) ) ) ) ).

% obtain_list_from_elements
tff(fact_281_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_re9121486324101951821t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: real] : pp(aa_nat_bool(aa_real_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_real] :
            ( ( size_size_list_real(L2) = N )
           => ~ ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
                 => pp(aa_nat_bool(aa_real_fun_nat_bool(P,aa_nat_real(nth_real(L2),I5)),I5)) ) ) ) ).

% obtain_list_from_elements
tff(fact_282_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_bo3288154511880838028t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: bool] : pp(aa_nat_bool(aa_bool_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_bool] :
            ( ( size_size_list_bool(L2) = N )
           => ~ ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
                 => pp(aa_nat_bool(aa_bool_fun_nat_bool(P,aa_nat_bool(nth_bool(L2),I5)),I5)) ) ) ) ).

% obtain_list_from_elements
tff(fact_283_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_nat] :
            ( ( size_size_list_nat(L2) = N )
           => ~ ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(L2),I5)),I5)) ) ) ) ).

% obtain_list_from_elements
tff(fact_284_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE3025410955681380100t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: vEBT_VEBTi] : pp(aa_nat_bool(aa_VEB8502397618584205677t_bool(P,Li),I2)) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L2) = N )
           => ~ ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
                 => pp(aa_nat_bool(aa_VEB8502397618584205677t_bool(P,nth_VEBT_VEBTi(L2,I5)),I5)) ) ) ) ).

% obtain_list_from_elements
tff(fact_285_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I4) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_286_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
           => ( aa_nat_real(nth_real(Xs),I4) = aa_nat_real(nth_real(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_287_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I4))
            <=> pp(aa_nat_bool(nth_bool(Ys),I4)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_288_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_289_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s7982070591426661849_VEBTi(Xs)))
           => ( nth_VEBT_VEBTi(Xs,I4) = nth_VEBT_VEBTi(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_290_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_291_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6801867027684862029l_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: real] : pp(aa_real_bool(aa_nat_fun_real_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_real] :
          ( ( size_size_list_real(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_real_bool(aa_nat_fun_real_bool(P,I4),aa_nat_real(nth_real(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_292_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),aa_nat_bool(nth_bool(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_293_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(nth_nat(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_294_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na307722919519257988i_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_1: vEBT_VEBTi] : pp(aa_VEBT_VEBTi_bool(aa_nat6176657665065580909i_bool(P,I4),X_1)) )
    <=> ? [Xs3: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_VEBT_VEBTi_bool(aa_nat6176657665065580909i_bool(P,I4),nth_VEBT_VEBTi(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_295_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_296_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
           => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_297_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Ys),I2)) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_298_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_299_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
           => ( nth_VEBT_VEBTi(Xs,I2) = nth_VEBT_VEBTi(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_300_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit0_div_2
tff(fact_301_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit0_div_2
tff(fact_302_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X2) = Xs )
      <=> ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_303_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( ( list_update_real(Xs,I,X2) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_304_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I,X2) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X2) ) ) ) ).

% list_update_same_conv
tff(fact_305_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X2) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_306_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,X2) = Xs )
      <=> ( nth_VEBT_VEBTi(Xs,I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_307_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))) )
       => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,X2)),J) = X2 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))) )
       => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,X2)),J) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),J) ) ) ) ).

% nth_list_update'
tff(fact_308_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_real,X2: real] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L))) )
       => ( aa_nat_real(nth_real(list_update_real(L,I,X2)),J) = X2 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L))) )
       => ( aa_nat_real(nth_real(list_update_real(L,I,X2)),J) = aa_nat_real(nth_real(L),J) ) ) ) ).

% nth_list_update'
tff(fact_309_nth__list__update_H,axiom,
    ! [L: list_bool,I: nat,X2: bool,J: nat] :
      ( pp(aa_nat_bool(nth_bool(list_update_bool(L,I,X2)),J))
    <=> ( ( ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L))) )
         => pp(X2) )
        & ( ~ ( ( I = J )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L))) )
         => pp(aa_nat_bool(nth_bool(L),J)) ) ) ) ).

% nth_list_update'
tff(fact_310_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X2: nat] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X2)),J) = X2 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X2)),J) = aa_nat_nat(nth_nat(L),J) ) ) ) ).

% nth_list_update'
tff(fact_311_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))) )
       => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,X2),J) = X2 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))) )
       => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,X2),J) = nth_VEBT_VEBTi(L,J) ) ) ) ).

% nth_list_update'
tff(fact_312_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2)),J) = X2 ) )
        & ( ( I != J )
         => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2)),J) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_313_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,J: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_real(nth_real(list_update_real(Xs,I,X2)),J) = X2 ) )
        & ( ( I != J )
         => ( aa_nat_real(nth_real(list_update_real(Xs,I,X2)),J) = aa_nat_real(nth_real(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_314_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X2: bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X2)),J))
      <=> ( ( ( I = J )
           => pp(X2) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_315_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),J) = X2 ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_316_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2),J) = X2 ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X2),J) = nth_VEBT_VEBTi(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_317_highbound,axiom,
    ( ( ma != mi )
   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,xa),ma))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList))) ) ) ).

% highbound
tff(fact_318_xbound,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),xa))
   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,xa),ma))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList))) ) ) ).

% xbound
tff(fact_319_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% member_inv
tff(fact_320_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_321_pred__list__to__short,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_322_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_323_succ__min,axiom,
    ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_324_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_325_highboundn,axiom,
    ( ( ma != mi )
   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,xa),ma))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xnew),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList))) ) ) ).

% highboundn
tff(fact_326_less__option__None__Some__code,axiom,
    ! [X2: nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),some_nat(X2))) ).

% less_option_None_Some_code
tff(fact_327_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_328_less__option__Some,axiom,
    ! [X2: real,Y3: real] :
      ( ord_less_option_real(some_real(X2),some_real(Y3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ).

% less_option_Some
tff(fact_329_less__option__Some,axiom,
    ! [X2: rat,Y3: rat] :
      ( ord_less_option_rat(some_rat(X2),some_rat(Y3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ).

% less_option_Some
tff(fact_330_less__option__Some,axiom,
    ! [X2: num,Y3: num] :
      ( ord_less_option_num(some_num(X2),some_num(Y3))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3)) ) ).

% less_option_Some
tff(fact_331_less__option__Some,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,some_nat(X2)),some_nat(Y3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ).

% less_option_Some
tff(fact_332_less__option__Some,axiom,
    ! [X2: int,Y3: int] :
      ( ord_less_option_int(some_int(X2),some_int(Y3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3)) ) ).

% less_option_Some
tff(fact_333_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2)) ) ).

% min_Null_member
tff(fact_334_less__eq__option__None__code,axiom,
    ! [X2: option_nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,none_nat),X2)) ).

% less_eq_option_None_code
tff(fact_335_less__option__None,axiom,
    ! [X2: option_nat] : ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,X2),none_nat)) ).

% less_option_None
tff(fact_336_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_337_less__eq__option__Some,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ord_le353528952715127954et_int(some_set_int(X2),some_set_int(Y3))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3)) ) ).

% less_eq_option_Some
tff(fact_338_less__eq__option__Some,axiom,
    ! [X2: rat,Y3: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(X2),some_rat(Y3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% less_eq_option_Some
tff(fact_339_less__eq__option__Some,axiom,
    ! [X2: num,Y3: num] :
      ( ord_le6622620407824499402on_num(some_num(X2),some_num(Y3))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3)) ) ).

% less_eq_option_Some
tff(fact_340_less__eq__option__Some,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,some_nat(X2)),some_nat(Y3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ).

% less_eq_option_Some
tff(fact_341_less__eq__option__Some,axiom,
    ! [X2: int,Y3: int] :
      ( ord_le1736525451366464988on_int(some_int(X2),some_int(Y3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ).

% less_eq_option_Some
tff(fact_342_less__eq__option__Some__None,axiom,
    ! [X2: nat] : ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,some_nat(X2)),none_nat)) ).

% less_eq_option_Some_None
tff(fact_343__C7_Oprems_C,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),na)) ).

% "7.prems"
tff(fact_344_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M3),N3))
             => pp(aa_Ext2772979874139974381t_bool(P,M3)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N3)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).

% enat_less_induct
tff(fact_345_less__eq__option__None,axiom,
    ! [X2: option_nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,none_nat),X2)) ).

% less_eq_option_None
tff(fact_346_less__eq__option__None__is__None,axiom,
    ! [X2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,X2),none_nat))
     => ( X2 = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_347_le__num__One__iff,axiom,
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),one))
    <=> ( X2 = one ) ) ).

% le_num_One_iff
tff(fact_348_less__option__None__Some,axiom,
    ! [X2: nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),some_nat(X2))) ).

% less_option_None_Some
tff(fact_349_less__option__None__is__Some,axiom,
    ! [X2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),X2))
     => ? [Z2: nat] : X2 = some_nat(Z2) ) ).

% less_option_None_is_Some
tff(fact_350_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2))
    <=> ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_351_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X2 != Ma )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_352_insert__simp__norm,axiom,
    ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X2 != Ma )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_353_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_354_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_355_mimaxprop,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))) ) ).

% mimaxprop
tff(fact_356_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),X2)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary) ).

% vebt_insert.simps(4)
tff(fact_357_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,one),N)) ).

% semiring_norm(68)
tff(fact_358_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).

% semiring_norm(75)
tff(fact_359_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(71)
tff(fact_360_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(78)
tff(fact_361_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_362_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)),N)) ) ).

% delete_pres_valid
tff(fact_363_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2),S3) ) ).

% deg_SUcn_Node
tff(fact_364_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)),Y3))
      <=> ( ( X2 != Y3 )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y3)) ) ) ) ).

% dele_member_cont_corr
tff(fact_365_power__shift,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),Y3) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X2)),some_nat(Y3)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_366_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% mint_member
tff(fact_367_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_368_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X2)) ) ) ) ).

% mint_corr_help
tff(fact_369_mulcomm,axiom,
    ! [I: nat,Va: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),I) ).

% mulcomm
tff(fact_370_high__def,axiom,
    ! [X2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_371_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X2))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% member_bound
tff(fact_372_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Ma),X2))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_373_semiring__norm_I83_J,axiom,
    ! [N: num] : one != aa_num_num(bit0,N) ).

% semiring_norm(83)
tff(fact_374_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_375_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% misiz
tff(fact_376_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X2) = some_nat(Y3) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% helpypredd
tff(fact_377_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X2) = some_nat(Y3) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% helpyd
tff(fact_378_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X2)),Y3))
           => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y3))
              | ( X2 = Y3 ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_379_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_380_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa_num_num(times_times_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(times_times_num(M),N))) ).

% semiring_norm(13)
tff(fact_381_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(times_times_num(M),one) = M ).

% semiring_norm(11)
tff(fact_382_semiring__norm_I12_J,axiom,
    ! [N: num] : aa_num_num(times_times_num(one),N) = N ).

% semiring_norm(12)
tff(fact_383_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_384_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2))
      <=> pp(aa_set_nat_bool(member_nat(X2),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_385_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_386_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(1)
tff(fact_387_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(1)
tff(fact_388_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_389_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_390_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_max_right
tff(fact_391_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_max_left
tff(fact_392_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_393_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_394_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X2)) ).

% vebt_member.simps(2)
tff(fact_395_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)) ).

% vebt_insert.simps(5)
tff(fact_396_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(X2) )
       => pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X2)) ) ) ).

% mint_corr
tff(fact_397_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X2))
       => ( vEBT_vebt_mint(T) = some_nat(X2) ) ) ) ).

% mint_sound
tff(fact_398_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M5: nat] :
              ( ( some_nat(M5) = vEBT_vebt_mint(Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_399_power__mult__numeral,axiom,
    ! [A2: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N))) ).

% power_mult_numeral
tff(fact_400_power__mult__numeral,axiom,
    ! [A2: real,M: num,N: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N))) ).

% power_mult_numeral
tff(fact_401_power__mult__numeral,axiom,
    ! [A2: int,M: num,N: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N))) ).

% power_mult_numeral
tff(fact_402_power__mult__numeral,axiom,
    ! [A2: complex,M: num,N: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N))) ).

% power_mult_numeral
tff(fact_403_power__mult__numeral,axiom,
    ! [A2: code_integer,M: num,N: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N))) ).

% power_mult_numeral
tff(fact_404_listlength,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% listlength
tff(fact_405_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% two_pow_div_gt_le
tff(fact_406_sumprop,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sumprop
tff(fact_407_power__odd__eq,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_408_power__odd__eq,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_409_power__odd__eq,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_410_power__odd__eq,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_411_power__odd__eq,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_412_power__odd__eq,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_413_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% two_powr_height_bound_deg
tff(fact_414_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),X2)) ) ) ) ).

% both_member_options_ding
tff(fact_415_setprop,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T),set_VEBT_VEBT2(treeList)))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% setprop
tff(fact_416_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12)) ) ).

% not_min_Null_member
tff(fact_417_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T)) ).

% set_vebt'_def
tff(fact_418_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)),Y3))
      <=> ( ( X2 != Y3 )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),Y3)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_419_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2)) ) ) ).

% valid_member_both_member_options
tff(fact_420_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2)) ) ) ).

% both_member_options_equiv_member
tff(fact_421_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_422_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_423_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N))) ) ) ).

% inthall
tff(fact_424_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_425_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_bool_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_426_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_427_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool,N: nat] :
      ( ! [X4: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(Xs))
         => pp(aa_VEBT_VEBTi_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,N))) ) ) ).

% inthall
tff(fact_428_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X6: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X6),X_13)) )
          & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_429_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_430_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_431_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X2)),X2)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_432_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Y3)),X2)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_433_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_434_left__diff__distrib__numeral,axiom,
    ! [A2: complex,B2: complex,V: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_435_left__diff__distrib__numeral,axiom,
    ! [A2: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_436_left__diff__distrib__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B2),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_437_left__diff__distrib__numeral,axiom,
    ! [A2: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_438_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B2),C2)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),B2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_439_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_440_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_441_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_442_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J)),J,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_443_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_real(Xs)))
       => ( set_real2(list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J)),J,aa_nat_real(nth_real(Xs),I))) = set_real2(Xs) ) ) ) ).

% set_swap
tff(fact_444_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_445_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_446_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs)))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,J)),J,nth_VEBT_VEBTi(Xs,I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_447_succ__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X2),Y3))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
        & ! [Z3: nat] :
            ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Z3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),Z3)) ) ) ) ).

% succ_member
tff(fact_448_pred__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X2),Y3))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
        & ! [Z3: nat] :
            ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Z3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z3),Y3)) ) ) ) ).

% pred_member
tff(fact_449_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X2) = some_nat(Px) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X2),Px)) ) ) ).

% pred_corr
tff(fact_450_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X2) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X2),Sx)) ) ) ).

% succ_corr
tff(fact_451_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X2) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_set_vebt(T)),X2),Sx)) ) ) ).

% pred_correct
tff(fact_452_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X2) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_set_vebt(T)),X2),Sx)) ) ) ).

% succ_correct
tff(fact_453_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_454_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B))
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),B)) ) ) ).

% subset_code(1)
tff(fact_455_subset__code_I1_J,axiom,
    ! [Xs: list_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X3),B)) ) ) ).

% subset_code(1)
tff(fact_456_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),B))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X3),B)) ) ) ).

% subset_code(1)
tff(fact_457_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X3),B)) ) ) ).

% subset_code(1)
tff(fact_458_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_459_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P,N3)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_460_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_461_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_462_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_463_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ) ) ).

% le_diff_iff'
tff(fact_464_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

% diff_le_self
tff(fact_465_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_466_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_467_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_468_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_469_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% diff_mult_distrib
tff(fact_470_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% diff_mult_distrib2
tff(fact_471_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A: set_VEBT_VEBTi,X2: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A)
     => ( member_VEBT_VEBTi(X2,A)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,X2)),A) ) ) ).

% set_update_subsetI
tff(fact_472_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X2))),A)) ) ) ).

% set_update_subsetI
tff(fact_473_set__update__subsetI,axiom,
    ! [Xs: list_nat,A: set_nat,X2: nat,I: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),A))
     => ( pp(aa_set_nat_bool(member_nat(X2),A))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X2))),A)) ) ) ).

% set_update_subsetI
tff(fact_474_set__update__subsetI,axiom,
    ! [Xs: list_bool,A: set_bool,X2: bool,I: nat] :
      ( ord_less_eq_set_bool(set_bool2(Xs),A)
     => ( member_bool(X2,A)
       => ord_less_eq_set_bool(set_bool2(list_update_bool(Xs,I,X2)),A) ) ) ).

% set_update_subsetI
tff(fact_475_set__update__subsetI,axiom,
    ! [Xs: list_real,A: set_real,X2: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X2))),A)) ) ) ).

% set_update_subsetI
tff(fact_476_set__update__subsetI,axiom,
    ! [Xs: list_int,A: set_int,X2: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A))
     => ( pp(aa_set_int_bool(member_int(X2),A))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X2))),A)) ) ) ).

% set_update_subsetI
tff(fact_477_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_478_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_479_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_480_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2))) ) ) ).

% diff_less_mono
tff(fact_481_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_482_power2__commute,axiom,
    ! [X2: complex,Y3: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y3),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_483_power2__commute,axiom,
    ! [X2: code_integer,Y3: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Y3),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_484_power2__commute,axiom,
    ! [X2: real,Y3: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_485_power2__commute,axiom,
    ! [X2: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y3),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_486_power2__commute,axiom,
    ! [X2: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Y3),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_487_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_488_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_489_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).

% nth_mem
tff(fact_490_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(aa_nat_bool(nth_bool(Xs),N),set_bool2(Xs)) ) ).

% nth_mem
tff(fact_491_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_492_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => member_VEBT_VEBTi(nth_VEBT_VEBTi(Xs,N),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_493_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X4)) )
       => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_494_list__ball__nth,axiom,
    ! [N: nat,Xs: list_real,P: fun_real_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
           => pp(aa_real_bool(P,X4)) )
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_495_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( ! [X4: bool] :
            ( member_bool(X4,set_bool2(Xs))
           => pp(aa_bool_bool(P,X4)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_496_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X4)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_497_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( ! [X4: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(Xs))
           => pp(aa_VEBT_VEBTi_bool(P,X4)) )
       => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_498_in__set__conv__nth,axiom,
    ! [X2: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I4) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_499_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I4) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_500_in__set__conv__nth,axiom,
    ! [X2: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I4) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_501_in__set__conv__nth,axiom,
    ! [X2: bool,Xs: list_bool] :
      ( member_bool(X2,set_bool2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I4))
          <=> pp(X2) ) ) ) ).

% in_set_conv_nth
tff(fact_502_in__set__conv__nth,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I4) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_503_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s7982070591426661849_VEBTi(Xs)))
          & ( nth_VEBT_VEBTi(Xs,I4) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_504_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(L)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(L)))
         => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_505_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_bool] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(L)))
         => pp(aa_real_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(L)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_506_all__set__conv__nth,axiom,
    ! [L: list_bool,P: fun_bool_bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(L))
         => pp(aa_bool_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(L)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_507_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(L)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(L)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_508_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( ! [X3: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(L))
         => pp(aa_VEBT_VEBTi_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s7982070591426661849_VEBTi(L)))
         => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(L,I4))) ) ) ).

% all_set_conv_nth
tff(fact_509_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X2: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) )
     => ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
       => pp(aa_int_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_510_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X2: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2))) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_511_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X2: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) )
     => ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
       => pp(aa_real_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_512_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X2: bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) )
     => ( member_bool(X2,set_bool2(Xs))
       => pp(aa_bool_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_513_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X2: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) )
     => ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_514_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool,X2: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,I2))) )
     => ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
       => pp(aa_VEBT_VEBTi_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_515_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_516_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_bool] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
         => pp(aa_real_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_517_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => pp(aa_bool_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_518_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_519_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( ! [X3: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
         => pp(aa_VEBT_VEBTi_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_520_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X2: int,Y3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( pp(aa_set_int_bool(member_int(X2),set_int2(list_update_int(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ( pp(aa_set_int_bool(member_int(X2),set_int2(L)))
            & ! [Y: int] : pp(aa_set_int_bool(member_int(X2),set_int2(list_update_int(L,I,Y)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_521_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(L)))
            & ! [Y: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_522_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( pp(aa_set_real_bool(member_real(X2),set_real2(list_update_real(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ( pp(aa_set_real_bool(member_real(X2),set_real2(L)))
            & ! [Y: real] : pp(aa_set_real_bool(member_real(X2),set_real2(list_update_real(L,I,Y)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_523_in__set__upd__eq,axiom,
    ! [I: nat,L: list_bool,X2: bool,Y3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( member_bool(X2,set_bool2(list_update_bool(L,I,Y3)))
      <=> ( ( pp(X2)
          <=> pp(Y3) )
          | ( member_bool(X2,set_bool2(L))
            & ! [Y: bool] : member_bool(X2,set_bool2(list_update_bool(L,I,Y))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_524_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(list_update_nat(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(L)))
            & ! [Y: nat] : pp(aa_set_nat_bool(member_nat(X2),set_nat2(list_update_nat(L,I,Y)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_525_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi,Y3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3)))
      <=> ( ( X2 = Y3 )
          | ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(L))
            & ! [Y: vEBT_VEBTi] : member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_526_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(X2),set_int2(list_update_int(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_527_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_528_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(X2),set_real2(list_update_real(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_529_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(X2,set_bool2(list_update_bool(Xs,N,X2))) ) ).

% set_update_memI
tff(fact_530_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(X2),set_nat2(list_update_nat(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_531_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,N,X2))) ) ).

% set_update_memI
tff(fact_532_in__set__upd__cases,axiom,
    ! [X2: int,L: list_int,I: nat,Y3: int] :
      ( pp(aa_set_int_bool(member_int(X2),set_int2(list_update_int(L,I,Y3))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
         => ( X2 != Y3 ) )
       => pp(aa_set_int_bool(member_int(X2),set_int2(L))) ) ) ).

% in_set_upd_cases
tff(fact_533_in__set__upd__cases,axiom,
    ! [X2: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y3: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
         => ( X2 != Y3 ) )
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(L))) ) ) ).

% in_set_upd_cases
tff(fact_534_in__set__upd__cases,axiom,
    ! [X2: real,L: list_real,I: nat,Y3: real] :
      ( pp(aa_set_real_bool(member_real(X2),set_real2(list_update_real(L,I,Y3))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
         => ( X2 != Y3 ) )
       => pp(aa_set_real_bool(member_real(X2),set_real2(L))) ) ) ).

% in_set_upd_cases
tff(fact_535_in__set__upd__cases,axiom,
    ! [X2: bool,L: list_bool,I: nat,Y3: bool] :
      ( member_bool(X2,set_bool2(list_update_bool(L,I,Y3)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
         => ( pp(X2)
          <=> ~ pp(Y3) ) )
       => member_bool(X2,set_bool2(L)) ) ) ).

% in_set_upd_cases
tff(fact_536_in__set__upd__cases,axiom,
    ! [X2: nat,L: list_nat,I: nat,Y3: nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(list_update_nat(L,I,Y3))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
         => ( X2 != Y3 ) )
       => pp(aa_set_nat_bool(member_nat(X2),set_nat2(L))) ) ) ).

% in_set_upd_cases
tff(fact_537_in__set__upd__cases,axiom,
    ! [X2: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y3: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
         => ( X2 != Y3 ) )
       => member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_538_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X2: int,Y3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( pp(aa_set_int_bool(member_int(X2),set_int2(list_update_int(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ! [Y: int] : pp(aa_set_int_bool(member_int(X2),set_int2(list_update_int(L,I,Y)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_539_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ! [Y: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_540_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( pp(aa_set_real_bool(member_real(X2),set_real2(list_update_real(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ! [Y: real] : pp(aa_set_real_bool(member_real(X2),set_real2(list_update_real(L,I,Y)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_541_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_bool,X2: bool,Y3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( member_bool(X2,set_bool2(list_update_bool(L,I,Y3)))
      <=> ( ( pp(X2)
          <=> pp(Y3) )
          | ! [Y: bool] : member_bool(X2,set_bool2(list_update_bool(L,I,Y))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_542_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(list_update_nat(L,I,Y3))))
      <=> ( ( X2 = Y3 )
          | ! [Y: nat] : pp(aa_set_nat_bool(member_nat(X2),set_nat2(list_update_nat(L,I,Y)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_543_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi,Y3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3)))
      <=> ( ( X2 = Y3 )
          | ! [Y: vEBT_VEBTi] : member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_544_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).

% diff_le_diff_pow
tff(fact_545_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) ) ) ).

% power_minus_is_div
tff(fact_546_less__two__pow__divD,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% less_two_pow_divD
tff(fact_547_less__two__pow__divI,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))) ) ) ).

% less_two_pow_divI
tff(fact_548_power__commutes,axiom,
    ! [A2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_commutes
tff(fact_549_power__commutes,axiom,
    ! [A2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),A2) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_commutes
tff(fact_550_power__commutes,axiom,
    ! [A2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_commutes
tff(fact_551_power__commutes,axiom,
    ! [A2: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),N)),A2) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N)) ).

% power_commutes
tff(fact_552_power__commutes,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_commutes
tff(fact_553_power__commutes,axiom,
    ! [A2: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ).

% power_commutes
tff(fact_554_power__mult__distrib,axiom,
    ! [A2: complex,B2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B2),N)) ).

% power_mult_distrib
tff(fact_555_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),N) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ).

% power_mult_distrib
tff(fact_556_power__mult__distrib,axiom,
    ! [A2: real,B2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ).

% power_mult_distrib
tff(fact_557_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A2),B2)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ).

% power_mult_distrib
tff(fact_558_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ).

% power_mult_distrib
tff(fact_559_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)) ).

% power_mult_distrib
tff(fact_560_power__commuting__commutes,axiom,
    ! [X2: complex,Y3: complex,N: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),X2) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)) ) ) ).

% power_commuting_commutes
tff(fact_561_power__commuting__commutes,axiom,
    ! [X2: code_integer,Y3: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X2),Y3) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y3),X2) )
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)),Y3) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y3),aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)) ) ) ).

% power_commuting_commutes
tff(fact_562_power__commuting__commutes,axiom,
    ! [X2: real,Y3: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),X2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) ) ) ).

% power_commuting_commutes
tff(fact_563_power__commuting__commutes,axiom,
    ! [X2: rat,Y3: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X2),Y3) = aa_rat_rat(times_times_rat(Y3),X2) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),N)),Y3) = aa_rat_rat(times_times_rat(Y3),aa_nat_rat(power_power_rat(X2),N)) ) ) ).

% power_commuting_commutes
tff(fact_564_power__commuting__commutes,axiom,
    ! [X2: nat,Y3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),X2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)) ) ) ).

% power_commuting_commutes
tff(fact_565_power__commuting__commutes,axiom,
    ! [X2: int,Y3: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),X2) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)) ) ) ).

% power_commuting_commutes
tff(fact_566_power__divide,axiom,
    ! [A2: complex,B2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B2),N)) ).

% power_divide
tff(fact_567_power__divide,axiom,
    ! [A2: real,B2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ).

% power_divide
tff(fact_568_power__divide,axiom,
    ! [A2: rat,B2: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A2),B2)),N) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ).

% power_divide
tff(fact_569_power__mult,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),N) ).

% power_mult
tff(fact_570_power__mult,axiom,
    ! [A2: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),N) ).

% power_mult
tff(fact_571_power__mult,axiom,
    ! [A2: int,M: nat,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),N) ).

% power_mult
tff(fact_572_power__mult,axiom,
    ! [A2: complex,M: nat,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),N) ).

% power_mult
tff(fact_573_power__mult,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),N) ).

% power_mult
tff(fact_574_power__Suc,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_Suc
tff(fact_575_power__Suc,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_Suc
tff(fact_576_power__Suc,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_Suc
tff(fact_577_power__Suc,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N)) ).

% power_Suc
tff(fact_578_power__Suc,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_Suc
tff(fact_579_power__Suc,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ).

% power_Suc
tff(fact_580_power__Suc2,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),A2) ).

% power_Suc2
tff(fact_581_power__Suc2,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),A2) ).

% power_Suc2
tff(fact_582_power__Suc2,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),A2) ).

% power_Suc2
tff(fact_583_power__Suc2,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),N)),A2) ).

% power_Suc2
tff(fact_584_power__Suc2,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),A2) ).

% power_Suc2
tff(fact_585_power__Suc2,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),A2) ).

% power_Suc2
tff(fact_586_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),A2)) ).

% div_mult_le
tff(fact_587_power__numeral__even,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_588_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] : aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_589_power__numeral__even,axiom,
    ! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_590_power__numeral__even,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_591_power__numeral__even,axiom,
    ! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_592_power__numeral__even,axiom,
    ! [Z: int,W: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_593_power4__eq__xxxx,axiom,
    ! [X2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),X2)),X2)),X2) ).

% power4_eq_xxxx
tff(fact_594_power4__eq__xxxx,axiom,
    ! [X2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X2),X2)),X2)),X2) ).

% power4_eq_xxxx
tff(fact_595_power4__eq__xxxx,axiom,
    ! [X2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),X2)),X2) ).

% power4_eq_xxxx
tff(fact_596_power4__eq__xxxx,axiom,
    ! [X2: rat] : aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(X2),X2)),X2)),X2) ).

% power4_eq_xxxx
tff(fact_597_power4__eq__xxxx,axiom,
    ! [X2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),X2)),X2)),X2) ).

% power4_eq_xxxx
tff(fact_598_power4__eq__xxxx,axiom,
    ! [X2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),X2)),X2) ).

% power4_eq_xxxx
tff(fact_599_power2__eq__square,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),A2) ).

% power2_eq_square
tff(fact_600_power2__eq__square,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),A2) ).

% power2_eq_square
tff(fact_601_power2__eq__square,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2) ).

% power2_eq_square
tff(fact_602_power2__eq__square,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A2),A2) ).

% power2_eq_square
tff(fact_603_power2__eq__square,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),A2) ).

% power2_eq_square
tff(fact_604_power2__eq__square,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2) ).

% power2_eq_square
tff(fact_605_power__even__eq,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_606_power__even__eq,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_607_power__even__eq,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_608_power__even__eq,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_609_power__even__eq,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_610_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_611_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_612_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_613_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X2))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N))) ) ).

% in_children_def
tff(fact_614_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_615_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_616_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% nat_less_power_trans
tff(fact_617_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_618_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% nat_le_power_trans
tff(fact_619_nat__power__less__diff,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))) ) ).

% nat_power_less_diff
tff(fact_620_xndef,axiom,
    xnew = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ).

% xndef
tff(fact_621_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D2))),L) ).

% bit_concat_def
tff(fact_622_low__inv,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X2)),N) = X2 ) ) ).

% low_inv
tff(fact_623_high__inv,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X2)),N) = Y3 ) ) ).

% high_inv
tff(fact_624_even__odd__cases,axiom,
    ! [X2: nat] :
      ( ! [N3: nat] : X2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3)
     => ~ ! [N3: nat] : X2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) ) ).

% even_odd_cases
tff(fact_625_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summary))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)))) ).

% height_compose_summary
tff(fact_626_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ).

% pow_sum
tff(fact_627_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T),set_VEBT_VEBT2(TreeList)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)))) ) ).

% height_compose_child
tff(fact_628_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ).

% high_bound_aux
tff(fact_629_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_630_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_631_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_632_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_633_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_634_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_635_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_636_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_637_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_638_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_639_power__one,axiom,
    ! [N: nat] : aa_nat_assn(power_power_assn(one_one_assn),N) = one_one_assn ).

% power_one
tff(fact_640_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_641_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_642_power__one,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),N) = one_one_real ).

% power_one
tff(fact_643_power__one,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),N) = one_one_int ).

% power_one
tff(fact_644_power__one,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),N) = one_one_complex ).

% power_one
tff(fact_645_power__one,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ).

% power_one
tff(fact_646_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_647_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_648_power__one__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_649_power__one__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_650_power__one__right,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_651_power__one__right,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_652_power__one__right,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_653_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_654_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_655_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_656_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_657_minewdef,axiom,
    minew = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ).

% minewdef
tff(fact_658_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_659_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_660_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_661_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_662_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_663_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_664_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_real(numeral_numeral_real,N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_665_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_666_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_667_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_668_distrib__right__numeral,axiom,
    ! [A2: complex,B2: complex,V: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_669_distrib__right__numeral,axiom,
    ! [A2: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_670_distrib__right__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B2),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_671_distrib__right__numeral,axiom,
    ! [A2: nat,B2: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_672_distrib__right__numeral,axiom,
    ! [A2: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_673_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B2),C2)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),B2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),C2)) ).

% distrib_left_numeral
tff(fact_674_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C2)) ).

% distrib_left_numeral
tff(fact_675_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% distrib_left_numeral
tff(fact_676_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C2)) ).

% distrib_left_numeral
tff(fact_677_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% distrib_left_numeral
tff(fact_678_power__inject__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_679_power__inject__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_680_power__inject__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( ( aa_nat_rat(power_power_rat(A2),M) = aa_nat_rat(power_power_rat(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_681_power__inject__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_682_power__inject__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_683_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_684_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_685_max__0__1_I6_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X2),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X2) ).

% max_0_1(6)
tff(fact_686_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X2)),one_one_real) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(6)
tff(fact_687_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X2)),one_one_rat) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(6)
tff(fact_688_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X2)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(6)
tff(fact_689_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X2)),one_one_int) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(6)
tff(fact_690_max__0__1_I5_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X2)) = numera1916890842035813515d_enat(X2) ).

% max_0_1(5)
tff(fact_691_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,X2)) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(5)
tff(fact_692_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X2)) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(5)
tff(fact_693_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X2)) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(5)
tff(fact_694_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,X2)) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(5)
tff(fact_695_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_696_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_697_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_698_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_699_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_700_newnodedef,axiom,
    newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% newnodedef
tff(fact_701_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_702_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_703_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_704_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_705_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_706_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_707_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_708_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_709_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_710_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_711_aktnodedef,axiom,
    ( ( ma != mi )
   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,xa),ma))
     => ( aktnode = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% aktnodedef
tff(fact_712_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,X2: nat,Y3: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),X2)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ) ).

% power_strict_increasing_iff
tff(fact_713_power__strict__increasing__iff,axiom,
    ! [B2: real,X2: nat,Y3: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ) ).

% power_strict_increasing_iff
tff(fact_714_power__strict__increasing__iff,axiom,
    ! [B2: rat,X2: nat,Y3: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B2),X2)),aa_nat_rat(power_power_rat(B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ) ).

% power_strict_increasing_iff
tff(fact_715_power__strict__increasing__iff,axiom,
    ! [B2: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ) ).

% power_strict_increasing_iff
tff(fact_716_power__strict__increasing__iff,axiom,
    ! [B2: int,X2: nat,Y3: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ) ).

% power_strict_increasing_iff
tff(fact_717_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_718_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_719_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_720_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% nested_mint
tff(fact_721_one__add__one,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_722_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_723_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_724_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_725_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_726_power__increasing__iff,axiom,
    ! [B2: code_integer,X2: nat,Y3: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),X2)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ).

% power_increasing_iff
tff(fact_727_power__increasing__iff,axiom,
    ! [B2: real,X2: nat,Y3: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ).

% power_increasing_iff
tff(fact_728_power__increasing__iff,axiom,
    ! [B2: rat,X2: nat,Y3: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),X2)),aa_nat_rat(power_power_rat(B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ).

% power_increasing_iff
tff(fact_729_power__increasing__iff,axiom,
    ! [B2: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ).

% power_increasing_iff
tff(fact_730_power__increasing__iff,axiom,
    ! [B2: int,X2: nat,Y3: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),Y3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ).

% power_increasing_iff
tff(fact_731_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_732_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_733_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_734_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_735_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_736_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_737_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_738_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_739_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_740_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_741_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_742_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_743__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aktnode_O_A_I_092_060lbrakk_062ma_A_092_060noteq_062_Ami_059_Ax_A_092_060le_062_Ama_092_060rbrakk_062_A_092_060Longrightarrow_062_Aaktnode_A_061_AtreeList_A_B_Ahigh_A_I2_A_K_A2_A_094_A_Iva_Adiv_A2_J_A_K_Athe_A_Ivebt__mint_Asummary_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_J_A_ISuc_A_Iva_Adiv_A2_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Aktnode: vEBT_VEBT] :
        ~ ( ( ma != mi )
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,xa),ma))
           => ( Aktnode = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% \<open>\<And>thesis. (\<And>aktnode. (\<lbrakk>ma \<noteq> mi; x \<le> ma\<rbrakk> \<Longrightarrow> aktnode = treeList ! high (2 * 2 ^ (va div 2) * the (vebt_mint summary) + the (vebt_mint (treeList ! the (vebt_mint summary)))) (Suc (va div 2))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_744_is__num__normalize_I1_J,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% is_num_normalize(1)
tff(fact_745_is__num__normalize_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% is_num_normalize(1)
tff(fact_746_is__num__normalize_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% is_num_normalize(1)
tff(fact_747_subset__Collect__conv,axiom,
    ! [S: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S),collect_nat(P)))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),S))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% subset_Collect_conv
tff(fact_748_subset__Collect__conv,axiom,
    ! [S: set_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),collect_complex(P)))
    <=> ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),S))
         => pp(aa_complex_bool(P,X3)) ) ) ).

% subset_Collect_conv
tff(fact_749_subset__Collect__conv,axiom,
    ! [S: set_Pr958786334691620121nt_int,P: fun_Pr803113274115125473t_bool] :
      ( ord_le2843351958646193337nt_int(S,collec213857154873943460nt_int(P))
    <=> ! [X3: product_prod_int_int] :
          ( member5262025264175285858nt_int(X3,S)
         => pp(aa_Pro4501881432599481050t_bool(P,X3)) ) ) ).

% subset_Collect_conv
tff(fact_750_subset__Collect__conv,axiom,
    ! [S: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S),collect_int(P)))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),S))
         => pp(aa_int_bool(P,X3)) ) ) ).

% subset_Collect_conv
tff(fact_751_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),numera6690914467698888265omplex(X2)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(X2)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_752_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X2)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_753_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X2)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X2)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_754_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X2)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_755_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X2)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_756_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_757_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_758_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_759_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_760_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_761_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,A2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(suc,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_762_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_763_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_764_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_765_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_less_mono1
tff(fact_766_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_767_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_768_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_769_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_770_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_771_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_772_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_773_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_774_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_775_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_776_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_777_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_778_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_779_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

% le_add1
tff(fact_780_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% le_add2
tff(fact_781_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_782_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_783_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N3: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N3) ) ).

% le_Suc_ex
tff(fact_784_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_785_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_le_mono1
tff(fact_786_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_787_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_788_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_789_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_790_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

% diff_add_inverse
tff(fact_791_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_792_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_793_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_794_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% add_mult_distrib2
tff(fact_795_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% add_mult_distrib
tff(fact_796_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_797_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_798_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_799_option_Osel,axiom,
    ! [X22: nat] : the_nat(some_nat(X22)) = X22 ).

% option.sel
tff(fact_800_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_801_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_802_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ).

% nat_add_max_right
tff(fact_803_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q3)) ).

% nat_add_max_left
tff(fact_804_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_805_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_806_numeral__Bit0,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_Bit0
tff(fact_807_numeral__Bit0,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_Bit0
tff(fact_808_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_809_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_810_power__add,axiom,
    ! [A2: complex,M: nat,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_add
tff(fact_811_power__add,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_add
tff(fact_812_power__add,axiom,
    ! [A2: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_add
tff(fact_813_power__add,axiom,
    ! [A2: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)) ).

% power_add
tff(fact_814_power__add,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_add
tff(fact_815_power__add,axiom,
    ! [A2: int,M: nat,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ).

% power_add
tff(fact_816_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_817_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_818_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_819_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_820_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q4: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ) ).

% less_natE
tff(fact_821_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N))) ).

% one_le_numeral
tff(fact_822_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% one_le_numeral
tff(fact_823_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_824_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).

% one_le_numeral
tff(fact_825_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_826_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_827_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_828_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_829_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M5: nat,N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M5)),aa_nat_nat(F,N3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_830_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_831_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_832_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_833_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_834_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_835_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_836_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_837_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_838_one__le__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% one_le_power
tff(fact_839_one__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% one_le_power
tff(fact_840_one__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% one_le_power
tff(fact_841_one__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% one_le_power
tff(fact_842_one__le__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ).

% one_le_power
tff(fact_843_mlex__bound,axiom,
    ! [A2: nat,A: nat,B2: nat,N6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N6))) ) ) ).

% mlex_bound
tff(fact_844_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,N6: nat,B4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),N6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B4),N6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N6)),B4))) ) ) ) ).

% mlex_fst_decrI
tff(fact_845_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B4: nat,N6: nat] :
      ( ( A2 = A4 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),B4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N6)),B4))) ) ) ).

% mlex_snd_decrI
tff(fact_846_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_847_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_848_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_849_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_850_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_851_left__right__inverse__power,axiom,
    ! [X2: assn,Y3: assn,N: nat] :
      ( ( aa_assn_assn(times_times_assn(X2),Y3) = one_one_assn )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(X2),N)),aa_nat_assn(power_power_assn(Y3),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_852_left__right__inverse__power,axiom,
    ! [X2: complex,Y3: complex,N: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Y3) = one_one_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_853_left__right__inverse__power,axiom,
    ! [X2: code_integer,Y3: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X2),Y3) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_854_left__right__inverse__power,axiom,
    ! [X2: real,Y3: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_855_left__right__inverse__power,axiom,
    ! [X2: rat,Y3: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X2),Y3) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),N)),aa_nat_rat(power_power_rat(Y3),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_856_left__right__inverse__power,axiom,
    ! [X2: nat,Y3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y3) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_857_left__right__inverse__power,axiom,
    ! [X2: int,Y3: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y3) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_858_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B4: nat,N6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),B4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N6)),B4))) ) ) ).

% mlex_leI
tff(fact_859_power__one__over,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),A2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_one_over
tff(fact_860_power__one__over,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_one_over
tff(fact_861_power__one__over,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A2)),N) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A2),N)) ).

% power_one_over
tff(fact_862_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_863_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_864_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_865_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_866_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_867_numeral__code_I2_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_code(2)
tff(fact_868_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_code(2)
tff(fact_869_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_code(2)
tff(fact_870_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_871_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_872_power__2__mult__step__le,axiom,
    ! [N2: nat,N: nat,K4: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),K4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K4),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K))) ) ) ).

% power_2_mult_step_le
tff(fact_873_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N3)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_874_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N3)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_875_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_876_power__less__power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))) ) ).

% power_less_power_Suc
tff(fact_877_power__less__power__Suc,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))) ) ).

% power_less_power_Suc
tff(fact_878_power__less__power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N)))) ) ).

% power_less_power_Suc
tff(fact_879_power__less__power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))) ) ).

% power_less_power_Suc
tff(fact_880_power__less__power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))) ) ).

% power_less_power_Suc
tff(fact_881_power__gt1__lemma,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))) ) ).

% power_gt1_lemma
tff(fact_882_power__gt1__lemma,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))) ) ).

% power_gt1_lemma
tff(fact_883_power__gt1__lemma,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N)))) ) ).

% power_gt1_lemma
tff(fact_884_power__gt1__lemma,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))) ) ).

% power_gt1_lemma
tff(fact_885_power__gt1__lemma,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))) ) ).

% power_gt1_lemma
tff(fact_886_power__gt1,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_887_power__gt1,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_888_power__gt1,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_889_power__gt1,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_890_power__gt1,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_891_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N6))) ) ) ).

% power_strict_increasing
tff(fact_892_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N6))) ) ) ).

% power_strict_increasing
tff(fact_893_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N6))) ) ) ).

% power_strict_increasing
tff(fact_894_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N6))) ) ) ).

% power_strict_increasing
tff(fact_895_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N6))) ) ) ).

% power_strict_increasing
tff(fact_896_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_897_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_898_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_899_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_900_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_901_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N6))) ) ) ).

% power_increasing
tff(fact_902_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N6))) ) ) ).

% power_increasing
tff(fact_903_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N6))) ) ) ).

% power_increasing
tff(fact_904_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N6))) ) ) ).

% power_increasing
tff(fact_905_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N6))) ) ) ).

% power_increasing
tff(fact_906_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_907_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_908_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_909_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_910_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_911_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_912_left__add__twice,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_913_left__add__twice,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_914_left__add__twice,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_915_left__add__twice,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_916_left__add__twice,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_917_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),Z) ).

% mult_2_right
tff(fact_918_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_919_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_920_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_921_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_922_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),Z) ).

% mult_2
tff(fact_923_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_924_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_925_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_926_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_927_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_928_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_929_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_930_one__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_931_one__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_932_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_power2
tff(fact_933_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_934_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_935_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_936_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_937_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_938_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_939_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_940_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_941_nat__add__offset__less,axiom,
    ! [Y3: nat,N: nat,X2: nat,M: nat,Sz: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Sz))) ) ) ) ).

% nat_add_offset_less
tff(fact_942_power2__sum,axiom,
    ! [X2: code_integer,Y3: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_sum
tff(fact_943_power2__sum,axiom,
    ! [X2: complex,Y3: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_sum
tff(fact_944_power2__sum,axiom,
    ! [X2: real,Y3: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_sum
tff(fact_945_power2__sum,axiom,
    ! [X2: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_sum
tff(fact_946_power2__sum,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_sum
tff(fact_947_power2__sum,axiom,
    ! [X2: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_sum
tff(fact_948_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N)) ) ) ).

% diff_diff_less
tff(fact_949_power2__diff,axiom,
    ! [X2: code_integer,Y3: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_diff
tff(fact_950_power2__diff,axiom,
    ! [X2: complex,Y3: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_diff
tff(fact_951_power2__diff,axiom,
    ! [X2: real,Y3: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_diff
tff(fact_952_power2__diff,axiom,
    ! [X2: rat,Y3: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_diff
tff(fact_953_power2__diff,axiom,
    ! [X2: int,Y3: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2)),Y3)) ).

% power2_diff
tff(fact_954_n__less__equal__power__2,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% n_less_equal_power_2
tff(fact_955_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),X2)),one_one_nat)) ) ).

% tdeletemimi'
tff(fact_956_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),X_1))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_957_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),X_1))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_958_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),X2)),one_one_nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_959_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% delete_bound_height'
tff(fact_960_del__x__mia,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = 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un_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_961_del__x__mi__lets__in__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H) )
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(Sn)),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(Sn)))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_962_del__x__mi__lets__in,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,Newlist),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H)) ) )
                        & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_963_del__x__mi,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X2 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_964_del__in__range,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_veb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aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)) ) ) ) ) ).

% del_in_range
tff(fact_965_del__x__not__mia,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
               => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_966_maxbmo,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(X2) )
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2)) ) ).

% maxbmo
tff(fact_967_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% maxt_member
tff(fact_968_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_969_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_970_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(X2) )
       => pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X2)) ) ) ).

% maxt_corr
tff(fact_971_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X2))
       => ( vEBT_vebt_maxt(T) = some_nat(X2) ) ) ) ).

% maxt_sound
tff(fact_972_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),N)) ).

% semiring_norm(6)
tff(fact_973_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_974_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_975_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% Suc_numeral
tff(fact_976_power__add__numeral,axiom,
    ! [A2: complex,M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).

% power_add_numeral
tff(fact_977_power__add__numeral,axiom,
    ! [A2: code_integer,M: num,N: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).

% power_add_numeral
tff(fact_978_power__add__numeral,axiom,
    ! [A2: real,M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).

% power_add_numeral
tff(fact_979_power__add__numeral,axiom,
    ! [A2: rat,M: num,N: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).

% power_add_numeral
tff(fact_980_power__add__numeral,axiom,
    ! [A2: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).

% power_add_numeral
tff(fact_981_power__add__numeral,axiom,
    ! [A2: int,M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).

% power_add_numeral
tff(fact_982_power__add__numeral2,axiom,
    ! [A2: complex,M: num,N: num,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B2) ).

% power_add_numeral2
tff(fact_983_power__add__numeral2,axiom,
    ! [A2: code_integer,M: num,N: num,B2: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,M))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B2) ).

% power_add_numeral2
tff(fact_984_power__add__numeral2,axiom,
    ! [A2: real,M: num,N: num,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B2) ).

% power_add_numeral2
tff(fact_985_power__add__numeral2,axiom,
    ! [A2: rat,M: num,N: num,B2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B2) ).

% power_add_numeral2
tff(fact_986_power__add__numeral2,axiom,
    ! [A2: nat,M: num,N: num,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B2) ).

% power_add_numeral2
tff(fact_987_power__add__numeral2,axiom,
    ! [A2: int,M: num,N: num,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B2) ).

% power_add_numeral2
tff(fact_988_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
               => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_989_del__x__mi__lets__in__not__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_990_del__x__not__mi,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,Newlist),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H)) ) )
                      & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_991_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
               => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(Sn)),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(Sn)))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_992_z1pdiv2,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = B2 ).

% z1pdiv2
tff(fact_993_add__One__commute,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),N) = aa_num_num(plus_plus_num(N),one) ).

% add_One_commute
tff(fact_994_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y3))
     => ( plus_p3455044024723400733d_enat(X2,minus_3235023915231533773d_enat(Y3,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X2,Y3),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_995_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_996_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_997_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_998_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_999_vebt__delete_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)) )
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary) ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(b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,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_1000_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)) )
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = one_one_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = one_one_nat ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_1001_sum__squares__bound,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)),Y3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_1002_sum__squares__bound,axiom,
    ! [X2: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X2)),Y3)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_1003_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)),one_one_real)))) ) ).

% cnt_bound'
tff(fact_1004__C7_OIH_C_I2_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,Xe: vEBT_VEBT,Xf: list_VEBT_VEBT,N: nat,Ti: vEBT_VEBTi] :
      ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),xa)) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( ( ( xa = mi )
             => ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ) )
            & ( ( xa != mi )
             => ( Xa = xa ) ) )
         => ( ( ( ( xa = mi )
               => ( Xb = Xa ) )
              & ( ( xa != mi )
               => ( Xb = mi ) ) )
           => ( ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
             => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xd),size_s6755466524823107622T_VEBT(treeList)))
                 => ( ( Xe = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Xc) )
                   => ( ( Xf = list_u1324408373059187874T_VEBT(treeList,Xd,Xe) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Xe))
                       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,summary),N))
                         => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),Ti),vEBT_V1365221501068881998eletei(summary,Ti,Xd),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(summary),Xd))) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(2)
tff(fact_1005_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)))) ) ).

% height_node
tff(fact_1006_le__add__diff__inverse,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1007_le__add__diff__inverse,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(minus_minus_rat(A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1008_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1009_le__add__diff__inverse,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1010_le__add__diff__inverse2,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1011_le__add__diff__inverse2,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1012_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1013_le__add__diff__inverse2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1014__C7_OIH_C_I1_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,N: nat,Ti: vEBT_VEBTi] :
      ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),xa)) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( ( ( xa = mi )
             => ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ) )
            & ( ( xa != mi )
             => ( Xa = xa ) ) )
         => ( ( ( ( xa = mi )
               => ( Xb = Xa ) )
              & ( ( xa != mi )
               => ( Xb = mi ) ) )
           => ( ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
             => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xd),size_s6755466524823107622T_VEBT(treeList)))
                 => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),N))
                   => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Ti),vEBT_V1365221501068881998eletei(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd),Ti,Xc),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Xc))) ) ) ) ) ) ) ) ) ).

% "7.IH"(1)
tff(fact_1015_div__exp__eq,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1016_div__exp__eq,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1017_div__exp__eq,axiom,
    ! [A2: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1018_bits__div__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),one_one_nat) = A2 ).

% bits_div_by_1
tff(fact_1019_bits__div__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),one_one_int) = A2 ).

% bits_div_by_1
tff(fact_1020_div__by__1,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),one_one_complex) = A2 ).

% div_by_1
tff(fact_1021_div__by__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),one_one_real) = A2 ).

% div_by_1
tff(fact_1022_div__by__1,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),one_one_rat) = A2 ).

% div_by_1
tff(fact_1023_div__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),one_one_nat) = A2 ).

% div_by_1
tff(fact_1024_div__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),one_one_int) = A2 ).

% div_by_1
tff(fact_1025_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_V3964819847710782039nserti(T,Ti,X2),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X2))) ).

% vebt_inserti'_rf_abstr
tff(fact_1026_linorder__neqE__linordered__idom,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1027_linorder__neqE__linordered__idom,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1028_linorder__neqE__linordered__idom,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1029_combine__common__factor,axiom,
    ! [A2: real,E: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1030_combine__common__factor,axiom,
    ! [A2: rat,E: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1031_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),E)),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1032_combine__common__factor,axiom,
    ! [A2: int,E: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1033_distrib__right,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% distrib_right
tff(fact_1034_distrib__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% distrib_right
tff(fact_1035_distrib__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% distrib_right
tff(fact_1036_distrib__right,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% distrib_right
tff(fact_1037_distrib__left,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% distrib_left
tff(fact_1038_distrib__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% distrib_left
tff(fact_1039_distrib__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)) ).

% distrib_left
tff(fact_1040_distrib__left,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)) ).

% distrib_left
tff(fact_1041_comm__semiring__class_Odistrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1042_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1043_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1044_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1045_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1046_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1047_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1048_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1049_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1050_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1051_right__diff__distrib_H,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% right_diff_distrib'
tff(fact_1052_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% right_diff_distrib'
tff(fact_1053_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)) ).

% right_diff_distrib'
tff(fact_1054_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)) ).

% right_diff_distrib'
tff(fact_1055_left__diff__distrib_H,axiom,
    ! [B2: real,C2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)) ).

% left_diff_distrib'
tff(fact_1056_left__diff__distrib_H,axiom,
    ! [B2: rat,C2: rat,A2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),C2)),A2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B2),A2)),aa_rat_rat(times_times_rat(C2),A2)) ).

% left_diff_distrib'
tff(fact_1057_left__diff__distrib_H,axiom,
    ! [B2: nat,C2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)) ).

% left_diff_distrib'
tff(fact_1058_left__diff__distrib_H,axiom,
    ! [B2: int,C2: int,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)) ).

% left_diff_distrib'
tff(fact_1059_right__diff__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% right_diff_distrib
tff(fact_1060_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% right_diff_distrib
tff(fact_1061_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)) ).

% right_diff_distrib
tff(fact_1062_left__diff__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% left_diff_distrib
tff(fact_1063_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% left_diff_distrib
tff(fact_1064_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% left_diff_distrib
tff(fact_1065_lambda__one,axiom,
    aTP_Lamm_af = times_times_assn(one_one_assn) ).

% lambda_one
tff(fact_1066_lambda__one,axiom,
    aTP_Lamm_ag = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_1067_lambda__one,axiom,
    aTP_Lamm_ah = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_1068_lambda__one,axiom,
    aTP_Lamm_ai = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1069_lambda__one,axiom,
    aTP_Lamm_aj = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1070_less__add__one,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real))) ).

% less_add_one
tff(fact_1071_less__add__one,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(A2),one_one_rat))) ).

% less_add_one
tff(fact_1072_less__add__one,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat))) ).

% less_add_one
tff(fact_1073_less__add__one,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int))) ).

% less_add_one
tff(fact_1074_add__mono1,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),one_one_real))) ) ).

% add_mono1
tff(fact_1075_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_rat_rat(plus_plus_rat(B2),one_one_rat))) ) ).

% add_mono1
tff(fact_1076_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),one_one_nat))) ) ).

% add_mono1
tff(fact_1077_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int))) ) ).

% add_mono1
tff(fact_1078_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N))) ) ) ).

% less_1_mult
tff(fact_1079_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N))) ) ) ).

% less_1_mult
tff(fact_1080_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% less_1_mult
tff(fact_1081_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).

% less_1_mult
tff(fact_1082_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1083_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1084_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1085_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1086_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1087_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1088_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1089_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1090_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: real,B2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1091_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(minus_minus_rat(A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1092_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1093_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1094_four__x__squared,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_1095_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1096_eq__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_1097_eq__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_1098_eq__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_1099_eq__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2) )
    <=> ( C2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_1100_eq__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2) )
    <=> ( C2 = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_1101_eq__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2) )
    <=> ( C2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_1102_square__diff__square__factored,axiom,
    ! [X2: real,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)) ).

% square_diff_square_factored
tff(fact_1103_square__diff__square__factored,axiom,
    ! [X2: rat,Y3: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y3),Y3)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X2),Y3)),aa_rat_rat(minus_minus_rat(X2),Y3)) ).

% square_diff_square_factored
tff(fact_1104_square__diff__square__factored,axiom,
    ! [X2: int,Y3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)) ).

% square_diff_square_factored
tff(fact_1105_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ).

% vebt_delete.simps(4)
tff(fact_1106_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),E)),C2)),D2)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1107_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),E)),C2)),D2)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1108_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E)),C2)),D2)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1109_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),E)),D2))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1110_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),A2)),E)),D2))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1111_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E)),D2))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1112_less__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),E)),D2))) ) ).

% less_add_iff2
tff(fact_1113_less__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),A2)),E)),D2))) ) ).

% less_add_iff2
tff(fact_1114_less__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E)),D2))) ) ).

% less_add_iff2
tff(fact_1115_less__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),E)),C2)),D2)) ) ).

% less_add_iff1
tff(fact_1116_less__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),E)),C2)),D2)) ) ).

% less_add_iff1
tff(fact_1117_less__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E)),C2)),D2)) ) ).

% less_add_iff1
tff(fact_1118_square__diff__one__factored,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)) ).

% square_diff_one_factored
tff(fact_1119_square__diff__one__factored,axiom,
    ! [X2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),X2)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X2),one_one_rat)),aa_rat_rat(minus_minus_rat(X2),one_one_rat)) ).

% square_diff_one_factored
tff(fact_1120_square__diff__one__factored,axiom,
    ! [X2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),one_one_int)) ).

% square_diff_one_factored
tff(fact_1121_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_1122__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__deletei_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060vebt__assn__raw_A_Ivebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: bool,X222: bool] :
      ( ( tia = vEBT_Leafi(X21,X222) )
     => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),tia),vEBT_V1365221501068881998eletei(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary),tia,xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),xa))) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_deletei' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <vebt_assn_raw (vebt_delete (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
tff(fact_1123_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1124_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1125_count__buildup,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)))) ).

% count_buildup
tff(fact_1126_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1127_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))))) ) ).

% cnt_bound
tff(fact_1128_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_1129_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_1130_field__less__half__sum,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_1131_field__less__half__sum,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),Y3)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_1132_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X2) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ak(T),X2)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_1133_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1134_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_1135_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1136_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1137_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X2) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al(T),X2)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_1138_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).

% semiring_norm(89)
tff(fact_1139_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_1140_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_1141_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_1142_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(80)
tff(fact_1143_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(73)
tff(fact_1144_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),N)) ).

% semiring_norm(7)
tff(fact_1145_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),N)) ).

% semiring_norm(9)
tff(fact_1146_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa_num_num(times_times_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(times_times_num(M),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_1147_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(times_times_num(aa_num_num(bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_1148_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(72)
tff(fact_1149_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(81)
tff(fact_1150_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_1151_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_1152_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_1153_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),N)),one)) ).

% semiring_norm(10)
tff(fact_1154_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),one)) ).

% semiring_norm(8)
tff(fact_1155_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_1156_semiring__norm_I4_J,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(plus_plus_num(N),one)) ).

% semiring_norm(4)
tff(fact_1157_semiring__norm_I3_J,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_1158_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),N)),aa_num_num(bit0,aa_num_num(times_times_num(M),N)))) ).

% semiring_norm(16)
tff(fact_1159_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(79)
tff(fact_1160_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(74)
tff(fact_1161_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_1162_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1163_set__notEmptyE,axiom,
    ! [S: set_VEBT_VEBT] :
      ( ( S != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X4: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S)) ) ).

% set_notEmptyE
tff(fact_1164_set__notEmptyE,axiom,
    ! [S: set_real] :
      ( ( S != bot_bot_set_real )
     => ~ ! [X4: real] : ~ pp(aa_set_real_bool(member_real(X4),S)) ) ).

% set_notEmptyE
tff(fact_1165_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X4: nat] : ~ pp(aa_set_nat_bool(member_nat(X4),S)) ) ).

% set_notEmptyE
tff(fact_1166_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X4: int] : ~ pp(aa_set_int_bool(member_int(X4),S)) ) ).

% set_notEmptyE
tff(fact_1167_memb__imp__not__empty,axiom,
    ! [X2: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),S))
     => ( S != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
tff(fact_1168_memb__imp__not__empty,axiom,
    ! [X2: real,S: set_real] :
      ( pp(aa_set_real_bool(member_real(X2),S))
     => ( S != bot_bot_set_real ) ) ).

% memb_imp_not_empty
tff(fact_1169_memb__imp__not__empty,axiom,
    ! [X2: nat,S: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),S))
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_1170_memb__imp__not__empty,axiom,
    ! [X2: int,S: set_int] :
      ( pp(aa_set_int_bool(member_int(X2),S))
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_1171_subset__minus__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( minus_minus_set_nat(A,B) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_1172_subset__minus__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( minus_minus_set_int(A,B) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_1173_num_Oexhaust,axiom,
    ! [Y3: num] :
      ( ( Y3 != one )
     => ( ! [X23: num] : Y3 != aa_num_num(bit0,X23)
       => ~ ! [X32: num] : Y3 != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_1174_xor__num_Ocases,axiom,
    ! [X2: product_prod_num_num] :
      ( ( X2 != product_Pair_num_num(one,one) )
     => ( ! [N3: num] : X2 != product_Pair_num_num(one,aa_num_num(bit0,N3))
       => ( ! [N3: num] : X2 != product_Pair_num_num(one,aa_num_num(bit1,N3))
         => ( ! [M5: num] : X2 != product_Pair_num_num(aa_num_num(bit0,M5),one)
           => ( ! [M5: num,N3: num] : X2 != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N3))
             => ( ! [M5: num,N3: num] : X2 != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N3))
               => ( ! [M5: num] : X2 != product_Pair_num_num(aa_num_num(bit1,M5),one)
                 => ( ! [M5: num,N3: num] : X2 != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N3))
                   => ~ ! [M5: num,N3: num] : X2 != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N3)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1175_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_1176_numeral__Bit1,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_Bit1
tff(fact_1177_numeral__Bit1,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_Bit1
tff(fact_1178_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_1179_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_1180_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_1181_less__eq__real__def,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% less_eq_real_def
tff(fact_1182_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X6: real] : pp(aa_set_real_bool(member_real(X6),S))
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z4)) )
       => ? [Y4: real] :
            ( ! [X6: real] :
                ( pp(aa_set_real_bool(member_real(X6),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X6),Y4)) )
            & ! [Z4: real] :
                ( ! [X4: real] :
                    ( pp(aa_set_real_bool(member_real(X4),S))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z4)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),Z4)) ) ) ) ) ).

% complete_real
tff(fact_1183_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_1184_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_code(3)
tff(fact_1185_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_code(3)
tff(fact_1186_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_1187_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_code(3)
tff(fact_1188_power__numeral__odd,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_1189_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] : aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_1190_power__numeral__odd,axiom,
    ! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_1191_power__numeral__odd,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_1192_power__numeral__odd,axiom,
    ! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_1193_power__numeral__odd,axiom,
    ! [Z: int,W: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_1194_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit1_div_2
tff(fact_1195_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit1_div_2
tff(fact_1196_power3__eq__cube,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_1197_power3__eq__cube,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_1198_power3__eq__cube,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_1199_power3__eq__cube,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_1200_power3__eq__cube,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_1201_power3__eq__cube,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_1202_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_1203_mult__commute__abs,axiom,
    ! [C2: real] : aa_rea7340605660300375871l_real(aTP_Lamm_am,C2) = aa_rea7340605660300375871l_real(times_times_real,C2) ).

% mult_commute_abs
tff(fact_1204_mult__commute__abs,axiom,
    ! [C2: rat] : aTP_Lamm_an(C2) = times_times_rat(C2) ).

% mult_commute_abs
tff(fact_1205_mult__commute__abs,axiom,
    ! [C2: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ao,C2) = aa_nat_fun_nat_nat(times_times_nat,C2) ).

% mult_commute_abs
tff(fact_1206_mult__commute__abs,axiom,
    ! [C2: int] : aa_int_fun_int_int(aTP_Lamm_ap,C2) = aa_int_fun_int_int(times_times_int,C2) ).

% mult_commute_abs
tff(fact_1207_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_1208_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),X2) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1209_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),Y3)),one_one_nat)) ) ) ).

% insersimp'
tff(fact_1210_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),L)),one_one_nat)) ) ) ).

% insertsimp'
tff(fact_1211_add__diff__add,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2)) ).

% add_diff_add
tff(fact_1212_add__diff__add,axiom,
    ! [A2: rat,C2: rat,B2: rat,D2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),aa_rat_rat(minus_minus_rat(C2),D2)) ).

% add_diff_add
tff(fact_1213_add__diff__add,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2)) ).

% add_diff_add
tff(fact_1214_real__arch__pow,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N3))) ) ).

% real_arch_pow
tff(fact_1215_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% insert'_bound_height
tff(fact_1216_small__powers__of__2,axiom,
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),one_one_nat)))) ) ).

% small_powers_of_2
tff(fact_1217_mult__diff__mult,axiom,
    ! [X2: real,Y3: real,A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),B2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),A2)),B2)) ).

% mult_diff_mult
tff(fact_1218_mult__diff__mult,axiom,
    ! [X2: rat,Y3: rat,A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),Y3)),aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),aa_rat_rat(minus_minus_rat(Y3),B2))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),A2)),B2)) ).

% mult_diff_mult
tff(fact_1219_mult__diff__mult,axiom,
    ! [X2: int,Y3: int,A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y3)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y3),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),A2)),B2)) ).

% mult_diff_mult
tff(fact_1220_discrete,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B2)) ) ).

% discrete
tff(fact_1221_discrete,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),B2)) ) ).

% discrete
tff(fact_1222_field__sum__of__halves,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X2 ).

% field_sum_of_halves
tff(fact_1223_field__sum__of__halves,axiom,
    ! [X2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X2 ).

% field_sum_of_halves
tff(fact_1224_space__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U))) ) ) ).

% space_bound
tff(fact_1225_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U))) ) ) ).

% space'_bound
tff(fact_1226_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ) ).

% delete_bound_height
tff(fact_1227_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ) ).

% minNull_delete_time_bound
tff(fact_1228_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% tdeletemimi
tff(fact_1229_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),U))) ) ).

% vebt_buildup_bound
tff(fact_1230_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U))) ) ).

% T_vebt_buildupi_univ
tff(fact_1231_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_1232_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_1233_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R2),numera6620942414471956472nteger(L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),R2) ) ) ) ).

% divmod_step_eq
tff(fact_1234_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Ti,X2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))))) ).

% TBOUND_vebt_inserti
tff(fact_1235_builupi_Hcorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))) ).

% builupi'corr
tff(fact_1236_space__space_H,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))) ).

% space_space'
tff(fact_1237_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V739175172307565963ildupi(N)),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% TBOUND_vebt_buildupi
tff(fact_1238_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Ti,X2)),one_one_nat)) ) ).

% TBOUND_minNull
tff(fact_1239_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_1240_TBOUND__replicate,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,C2: nat,N: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(X2),C2))
     => pp(aa_nat_bool(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(N,X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),N)))) ) ).

% TBOUND_replicate
tff(fact_1241_TBOUND__replicate,axiom,
    ! [X2: heap_Time_Heap_bool,C2: nat,N: nat] :
      ( pp(aa_nat_bool(time_TBOUND_bool(X2),C2))
     => pp(aa_nat_bool(time_T5840735555483777133t_bool(vEBT_V6044891547498605301i_bool(N,X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),N)))) ) ).

% TBOUND_replicate
tff(fact_1242_TBOUND__replicate,axiom,
    ! [X2: heap_T2636463487746394924on_nat,C2: nat,N: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(X2),C2))
     => pp(aa_nat_bool(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(N,X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),N)))) ) ).

% TBOUND_replicate
tff(fact_1243_TBOUND__replicate,axiom,
    ! [X2: heap_Time_Heap_nat,C2: nat,N: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(X2),C2))
     => pp(aa_nat_bool(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(N,X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),N)))) ) ).

% TBOUND_replicate
tff(fact_1244_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U))) ) ).

% htt_vebt_buildupi'_univ
tff(fact_1245_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Tb_T_vebt_buildupi''
tff(fact_1246_T__vebt__buildupi,axiom,
    ! [N: nat,H: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(N),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% T_vebt_buildupi
tff(fact_1247_builupicorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))) ).

% builupicorr
tff(fact_1248_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% htt_vebt_buildupi'
tff(fact_1249_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)),one_one_real)))) ) ).

% space_2_pow_bound
tff(fact_1250_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% zle_add1_eq_le
tff(fact_1251_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_1252_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat(A,B) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% Diff_eq_empty_iff
tff(fact_1253_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( minus_minus_set_int(A,B) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% Diff_eq_empty_iff
tff(fact_1254_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_set_nat_bool(member_nat(X4),B)) )
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% subsetI
tff(fact_1255_subsetI,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B)) ) ).

% subsetI
tff(fact_1256_subsetI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_set_real_bool(member_real(X4),B)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(A),B)) ) ).

% subsetI
tff(fact_1257_subsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_set_int_bool(member_int(X4),B)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% subsetI
tff(fact_1258_psubsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( A != B )
       => pp(aa_set_int_bool(ord_less_set_int(A),B)) ) ) ).

% psubsetI
tff(fact_1259_subset__antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
       => ( A = B ) ) ) ).

% subset_antisym
tff(fact_1260_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_1261_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_1262_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_1263_DiffI,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),A))
     => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),B))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B))) ) ) ).

% DiffI
tff(fact_1264_DiffI,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C2),A))
     => ( ~ pp(aa_set_real_bool(member_real(C2),B))
       => pp(aa_set_real_bool(member_real(C2),minus_minus_set_real(A,B))) ) ) ).

% DiffI
tff(fact_1265_DiffI,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C2),A))
     => ( ~ pp(aa_set_int_bool(member_int(C2),B))
       => pp(aa_set_int_bool(member_int(C2),minus_minus_set_int(A,B))) ) ) ).

% DiffI
tff(fact_1266_DiffI,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C2),A))
     => ( ~ pp(aa_set_nat_bool(member_nat(C2),B))
       => pp(aa_set_nat_bool(member_nat(C2),minus_minus_set_nat(A,B))) ) ) ).

% DiffI
tff(fact_1267_Diff__iff,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B)))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),A))
        & ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),B)) ) ) ).

% Diff_iff
tff(fact_1268_Diff__iff,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C2),minus_minus_set_real(A,B)))
    <=> ( pp(aa_set_real_bool(member_real(C2),A))
        & ~ pp(aa_set_real_bool(member_real(C2),B)) ) ) ).

% Diff_iff
tff(fact_1269_Diff__iff,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C2),minus_minus_set_int(A,B)))
    <=> ( pp(aa_set_int_bool(member_int(C2),A))
        & ~ pp(aa_set_int_bool(member_int(C2),B)) ) ) ).

% Diff_iff
tff(fact_1270_Diff__iff,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C2),minus_minus_set_nat(A,B)))
    <=> ( pp(aa_set_nat_bool(member_nat(C2),A))
        & ~ pp(aa_set_nat_bool(member_nat(C2),B)) ) ) ).

% Diff_iff
tff(fact_1271_Diff__idemp,axiom,
    ! [A: set_nat,B: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A,B),B) = minus_minus_set_nat(A,B) ).

% Diff_idemp
tff(fact_1272_time__replicate,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,C2: nat,N: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(X2,H2)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(N,X2),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),N)))) ) ).

% time_replicate
tff(fact_1273_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ) ).

% two_realpow_ge_two
tff(fact_1274_htt__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% htt_vebt_buildupi
tff(fact_1275_subset__empty,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1276_subset__empty,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1277_empty__subsetI,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(bot_bot_set_nat),A)) ).

% empty_subsetI
tff(fact_1278_empty__subsetI,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% empty_subsetI
tff(fact_1279_Diff__empty,axiom,
    ! [A: set_int] : minus_minus_set_int(A,bot_bot_set_int) = A ).

% Diff_empty
tff(fact_1280_Diff__empty,axiom,
    ! [A: set_nat] : minus_minus_set_nat(A,bot_bot_set_nat) = A ).

% Diff_empty
tff(fact_1281_empty__Diff,axiom,
    ! [A: set_int] : minus_minus_set_int(bot_bot_set_int,A) = bot_bot_set_int ).

% empty_Diff
tff(fact_1282_empty__Diff,axiom,
    ! [A: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1283_Diff__cancel,axiom,
    ! [A: set_int] : minus_minus_set_int(A,A) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1284_Diff__cancel,axiom,
    ! [A: set_nat] : minus_minus_set_nat(A,A) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1285_count__buildup_H,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))) ).

% count_buildup'
tff(fact_1286_space__cnt,axiom,
    ! [T: vEBT_VEBT] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)))) ).

% space_cnt
tff(fact_1287_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))))) ).

% T_vebt_buildupi_cnt'
tff(fact_1288_t__buildup__cnt,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ).

% t_buildup_cnt
tff(fact_1289_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).

% of_nat_numeral
tff(fact_1290_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% of_nat_numeral
tff(fact_1291_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = aa_num_real(numeral_numeral_real,N) ).

% of_nat_numeral
tff(fact_1292_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_1293_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_1294_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1295_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1296_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1297_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_1298_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1299_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1300_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1301_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1302_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_1303_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_1304_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_1305_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_1306_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_1307_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_1308_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_1309_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_1310_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1311_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1312_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1313_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_1314_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1315_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1316_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1317_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_1318_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_1319_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_1320_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_1321_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_1322_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,X2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B2)),W) )
    <=> ( X2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1323_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,X2) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W) )
    <=> ( X2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1324_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) )
    <=> ( X2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1325_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,X2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B2)),W) )
    <=> ( X2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1326_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,X2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) )
    <=> ( X2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1327_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B2)),W) = aa_nat_complex(semiri8010041392384452111omplex,X2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1328_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,X2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1329_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) = aa_nat_real(semiri5074537144036343181t_real,X2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1330_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B2)),W) = aa_nat_int(semiri1314217659103216013at_int,X2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1331_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1332_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_1333_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_1334_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_1335_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_1336_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_1337_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% zle_diff1_eq
tff(fact_1338_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U))) ) ).

% htt_vebt_buildupi_univ
tff(fact_1339_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_1340_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_1341_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_1342_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_1343_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1344_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X2)),N) = aa_nat_complex(semiri8010041392384452111omplex,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1345_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1346_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N) = aa_nat_real(semiri5074537144036343181t_real,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1347_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = aa_nat_int(semiri1314217659103216013at_int,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1348_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y3) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1349_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X2: num,N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y3) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N) )
    <=> ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1350_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X2: num,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y3) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X2)),N) )
    <=> ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1351_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X2: num,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y3) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N) )
    <=> ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1352_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X2: num,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N) )
    <=> ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1353_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X2: num,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) )
    <=> ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1354_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X2: num,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) )
    <=> ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1355_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1356_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1357_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1358_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1359_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1360_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1361_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1362_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1363_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1364_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1365_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1366_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1367_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1368_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1369_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B2: nat,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B2)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1370_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1371_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1372_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1373_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1374_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1375_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_1376_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_num_real(numeral_numeral_real,W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_1377_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1378_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1379_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1380_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1381_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1382_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1383_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1384_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1385_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1386_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1387_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1388_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1389_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1390_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1391_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1392_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1393_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1394_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1395_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1396_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1397_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1398_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : pp(aa_nat_bool(time_TBOUND_bool(vEBT_V854960066525838166emberi(T,Ti,X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_memberi
tff(fact_1399_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_predi(T,Ti,X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_predi
tff(fact_1400_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_succi(T,Ti,X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_succi
tff(fact_1401_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_inserti(Ti,X2),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% htt_vebt_inserti
tff(fact_1402_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X2)),Y3) = aa_rat_rat(times_times_rat(Y3),aa_nat_rat(semiri681578069525770553at_rat,X2)) ).

% mult_of_nat_commute
tff(fact_1403_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X2)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),aa_nat_real(semiri5074537144036343181t_real,X2)) ).

% mult_of_nat_commute
tff(fact_1404_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X2)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_nat_int(semiri1314217659103216013at_int,X2)) ).

% mult_of_nat_commute
tff(fact_1405_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y3),aa_nat_nat(semiri1316708129612266289at_nat,X2)) ).

% mult_of_nat_commute
tff(fact_1406_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_1407_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_1408_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_1409_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_1410_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1411_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1412_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1413_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_1414_div__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_1415_div__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% div_mult2_eq'
tff(fact_1416_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_1417_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_1418_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_1419_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_1420_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1421_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1422_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)) = ord_ma741700101516333627d_enat(semiri4216267220026989637d_enat(X2),semiri4216267220026989637d_enat(Y3)) ).

% of_nat_max
tff(fact_1423_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(semiri5074537144036343181t_real,Y3)) ).

% of_nat_max
tff(fact_1424_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Y3)) ).

% of_nat_max
tff(fact_1425_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(semiri1316708129612266289at_nat,Y3)) ).

% of_nat_max
tff(fact_1426_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_1427_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_1428_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_1429_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_1430_real__of__nat__div4,axiom,
    ! [N: nat,X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X2)))) ).

% real_of_nat_div4
tff(fact_1431_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(member_nat(X2),A))
       => pp(aa_set_nat_bool(member_nat(X2),B)) ) ) ).

% in_mono
tff(fact_1432_in__mono,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B)) ) ) ).

% in_mono
tff(fact_1433_in__mono,axiom,
    ! [A: set_real,B: set_real,X2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => pp(aa_set_real_bool(member_real(X2),B)) ) ) ).

% in_mono
tff(fact_1434_in__mono,axiom,
    ! [A: set_int,B: set_int,X2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(member_int(X2),A))
       => pp(aa_set_int_bool(member_int(X2),B)) ) ) ).

% in_mono
tff(fact_1435_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(member_nat(C2),A))
       => pp(aa_set_nat_bool(member_nat(C2),B)) ) ) ).

% subsetD
tff(fact_1436_subsetD,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),A))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),B)) ) ) ).

% subsetD
tff(fact_1437_subsetD,axiom,
    ! [A: set_real,B: set_real,C2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
     => ( pp(aa_set_real_bool(member_real(C2),A))
       => pp(aa_set_real_bool(member_real(C2),B)) ) ) ).

% subsetD
tff(fact_1438_subsetD,axiom,
    ! [A: set_int,B: set_int,C2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(member_int(C2),A))
       => pp(aa_set_int_bool(member_int(C2),B)) ) ) ).

% subsetD
tff(fact_1439_psubsetE,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
         => pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% psubsetE
tff(fact_1440_equalityE,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% equalityE
tff(fact_1441_subset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A))
         => pp(aa_set_nat_bool(member_nat(X3),B)) ) ) ).

% subset_eq
tff(fact_1442_subset__eq,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),B)) ) ) ).

% subset_eq
tff(fact_1443_subset__eq,axiom,
    ! [A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A))
         => pp(aa_set_real_bool(member_real(X3),B)) ) ) ).

% subset_eq
tff(fact_1444_subset__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A))
         => pp(aa_set_int_bool(member_int(X3),B)) ) ) ).

% subset_eq
tff(fact_1445_equalityD1,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% equalityD1
tff(fact_1446_Set_OequalityD2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ).

% Set.equalityD2
tff(fact_1447_psubset__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & ( A != B ) ) ) ).

% psubset_eq
tff(fact_1448_subset__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ! [T2: nat] :
          ( pp(aa_set_nat_bool(member_nat(T2),A))
         => pp(aa_set_nat_bool(member_nat(T2),B)) ) ) ).

% subset_iff
tff(fact_1449_subset__iff,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
    <=> ! [T2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T2),A))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T2),B)) ) ) ).

% subset_iff
tff(fact_1450_subset__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
    <=> ! [T2: real] :
          ( pp(aa_set_real_bool(member_real(T2),A))
         => pp(aa_set_real_bool(member_real(T2),B)) ) ) ).

% subset_iff
tff(fact_1451_subset__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ! [T2: int] :
          ( pp(aa_set_int_bool(member_int(T2),A))
         => pp(aa_set_int_bool(member_int(T2),B)) ) ) ).

% subset_iff
tff(fact_1452_subset__refl,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% subset_refl
tff(fact_1453_Collect__mono,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
         => pp(aa_nat_bool(Q,X4)) )
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))) ) ).

% Collect_mono
tff(fact_1454_Collect__mono,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X4: complex] :
          ( pp(aa_complex_bool(P,X4))
         => pp(aa_complex_bool(Q,X4)) )
     => pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q))) ) ).

% Collect_mono
tff(fact_1455_Collect__mono,axiom,
    ! [P: fun_Pr803113274115125473t_bool,Q: fun_Pr803113274115125473t_bool] :
      ( ! [X4: product_prod_int_int] :
          ( pp(aa_Pro4501881432599481050t_bool(P,X4))
         => pp(aa_Pro4501881432599481050t_bool(Q,X4)) )
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ) ).

% Collect_mono
tff(fact_1456_Collect__mono,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
         => pp(aa_int_bool(Q,X4)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q))) ) ).

% Collect_mono
tff(fact_1457_subset__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% subset_trans
tff(fact_1458_set__eq__subset,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% set_eq_subset
tff(fact_1459_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)))
    <=> ! [X3: nat] :
          ( pp(aa_nat_bool(P,X3))
         => pp(aa_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1460_Collect__mono__iff,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q)))
    <=> ! [X3: complex] :
          ( pp(aa_complex_bool(P,X3))
         => pp(aa_complex_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1461_Collect__mono__iff,axiom,
    ! [P: fun_Pr803113274115125473t_bool,Q: fun_Pr803113274115125473t_bool] :
      ( ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q))
    <=> ! [X3: product_prod_int_int] :
          ( pp(aa_Pro4501881432599481050t_bool(P,X3))
         => pp(aa_Pro4501881432599481050t_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1462_Collect__mono__iff,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q)))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(P,X3))
         => pp(aa_int_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1463_psubset__imp__subset,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% psubset_imp_subset
tff(fact_1464_psubset__subset__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% psubset_subset_trans
tff(fact_1465_subset__not__subset__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% subset_not_subset_eq
tff(fact_1466_subset__psubset__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% subset_psubset_trans
tff(fact_1467_subset__iff__psubset__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A),B))
        | ( A = B ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1468_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_1469_psubset__imp__ex__mem,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A),B))
     => ? [B3: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A))) ) ).

% psubset_imp_ex_mem
tff(fact_1470_psubset__imp__ex__mem,axiom,
    ! [A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A),B))
     => ? [B3: real] : pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A))) ) ).

% psubset_imp_ex_mem
tff(fact_1471_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => ? [B3: int] : pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B,A))) ) ).

% psubset_imp_ex_mem
tff(fact_1472_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => ? [B3: nat] : pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A))) ) ).

% psubset_imp_ex_mem
tff(fact_1473_DiffE,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B)))
     => ~ ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),A))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),B)) ) ) ).

% DiffE
tff(fact_1474_DiffE,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C2),minus_minus_set_real(A,B)))
     => ~ ( pp(aa_set_real_bool(member_real(C2),A))
         => pp(aa_set_real_bool(member_real(C2),B)) ) ) ).

% DiffE
tff(fact_1475_DiffE,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C2),minus_minus_set_int(A,B)))
     => ~ ( pp(aa_set_int_bool(member_int(C2),A))
         => pp(aa_set_int_bool(member_int(C2),B)) ) ) ).

% DiffE
tff(fact_1476_DiffE,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C2),minus_minus_set_nat(A,B)))
     => ~ ( pp(aa_set_nat_bool(member_nat(C2),A))
         => pp(aa_set_nat_bool(member_nat(C2),B)) ) ) ).

% DiffE
tff(fact_1477_DiffD1,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B)))
     => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),A)) ) ).

% DiffD1
tff(fact_1478_DiffD1,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C2),minus_minus_set_real(A,B)))
     => pp(aa_set_real_bool(member_real(C2),A)) ) ).

% DiffD1
tff(fact_1479_DiffD1,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C2),minus_minus_set_int(A,B)))
     => pp(aa_set_int_bool(member_int(C2),A)) ) ).

% DiffD1
tff(fact_1480_DiffD1,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C2),minus_minus_set_nat(A,B)))
     => pp(aa_set_nat_bool(member_nat(C2),A)) ) ).

% DiffD1
tff(fact_1481_DiffD2,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B)))
     => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C2),B)) ) ).

% DiffD2
tff(fact_1482_DiffD2,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(member_real(C2),minus_minus_set_real(A,B)))
     => ~ pp(aa_set_real_bool(member_real(C2),B)) ) ).

% DiffD2
tff(fact_1483_DiffD2,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(member_int(C2),minus_minus_set_int(A,B)))
     => ~ pp(aa_set_int_bool(member_int(C2),B)) ) ).

% DiffD2
tff(fact_1484_DiffD2,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C2),minus_minus_set_nat(A,B)))
     => ~ pp(aa_set_nat_bool(member_nat(C2),B)) ) ).

% DiffD2
tff(fact_1485_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_1486_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_1487_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_1488_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_1489_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_aq) ).

% Set.empty_def
tff(fact_1490_Set_Oempty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_ar) ).

% Set.empty_def
tff(fact_1491_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_as) ).

% Set.empty_def
tff(fact_1492_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_at) ).

% Set.empty_def
tff(fact_1493_Collect__subset,axiom,
    ! [A: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_au(A),P))),A)) ).

% Collect_subset
tff(fact_1494_Collect__subset,axiom,
    ! [A: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_av(A),P))),A)) ).

% Collect_subset
tff(fact_1495_Collect__subset,axiom,
    ! [A: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(A),P))),A)) ).

% Collect_subset
tff(fact_1496_Collect__subset,axiom,
    ! [A: set_complex,P: fun_complex_bool] : pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ax(A),P))),A)) ).

% Collect_subset
tff(fact_1497_Collect__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_Pr803113274115125473t_bool] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun8917503761599146496t_bool(aTP_Lamm_ay(A),P)),A) ).

% Collect_subset
tff(fact_1498_Collect__subset,axiom,
    ! [A: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_az(A),P))),A)) ).

% Collect_subset
tff(fact_1499_less__eq__set__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ord_le8840051076831633728t_bool(aTP_Lamm_ab(A),aTP_Lamm_ab(B)) ) ).

% less_eq_set_def
tff(fact_1500_less__eq__set__def,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
    <=> ord_le7893757273149057542T_bool(aTP_Lamm_a(A),aTP_Lamm_a(B)) ) ).

% less_eq_set_def
tff(fact_1501_less__eq__set__def,axiom,
    ! [A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
    <=> ord_le5573776112191778788l_bool(aTP_Lamm_aa(A),aTP_Lamm_aa(B)) ) ).

% less_eq_set_def
tff(fact_1502_less__eq__set__def,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ord_le5715760110363009508t_bool(aTP_Lamm_ac(A),aTP_Lamm_ac(B)) ) ).

% less_eq_set_def
tff(fact_1503_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_1504_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_1505_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_1506_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_1507_set__diff__eq,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,B) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_ba(A),B)) ).

% set_diff_eq
tff(fact_1508_set__diff__eq,axiom,
    ! [A: set_real,B: set_real] : minus_minus_set_real(A,B) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_bb(A),B)) ).

% set_diff_eq
tff(fact_1509_set__diff__eq,axiom,
    ! [A: set_int,B: set_int] : minus_minus_set_int(A,B) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_bc(A),B)) ).

% set_diff_eq
tff(fact_1510_set__diff__eq,axiom,
    ! [A: set_complex,B: set_complex] : minus_811609699411566653omplex(A,B) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_bd(A),B)) ).

% set_diff_eq
tff(fact_1511_set__diff__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A,B) = collec213857154873943460nt_int(aa_set1381271197253378494t_bool(aTP_Lamm_be(A),B)) ).

% set_diff_eq
tff(fact_1512_set__diff__eq,axiom,
    ! [A: set_nat,B: set_nat] : minus_minus_set_nat(A,B) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_bf(A),B)) ).

% set_diff_eq
tff(fact_1513_minus__set__def,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,B) = collect_VEBT_VEBT(minus_1873103217924207103T_bool(aTP_Lamm_a(A),aTP_Lamm_a(B))) ).

% minus_set_def
tff(fact_1514_minus__set__def,axiom,
    ! [A: set_real,B: set_real] : minus_minus_set_real(A,B) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_aa(A),aTP_Lamm_aa(B))) ).

% minus_set_def
tff(fact_1515_minus__set__def,axiom,
    ! [A: set_int,B: set_int] : minus_minus_set_int(A,B) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_ac(A),aTP_Lamm_ac(B))) ).

% minus_set_def
tff(fact_1516_minus__set__def,axiom,
    ! [A: set_complex,B: set_complex] : minus_811609699411566653omplex(A,B) = collect_complex(minus_5423061390740260265x_bool(aTP_Lamm_ad(A),aTP_Lamm_ad(B))) ).

% minus_set_def
tff(fact_1517_minus__set__def,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A,B) = collec213857154873943460nt_int(minus_7866439474815358650t_bool(aTP_Lamm_ae(A),aTP_Lamm_ae(B))) ).

% minus_set_def
tff(fact_1518_minus__set__def,axiom,
    ! [A: set_nat,B: set_nat] : minus_minus_set_nat(A,B) = collect_nat(minus_1628305797571225351t_bool(aTP_Lamm_ab(A),aTP_Lamm_ab(B))) ).

% minus_set_def
tff(fact_1519_real__of__nat__div3,axiom,
    ! [N: nat,X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X2))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X2)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_1520_double__diff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => ( minus_minus_set_nat(B,minus_minus_set_nat(C,A)) = A ) ) ) ).

% double_diff
tff(fact_1521_double__diff,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => ( minus_minus_set_int(B,minus_minus_set_int(C,A)) = A ) ) ) ).

% double_diff
tff(fact_1522_Diff__subset,axiom,
    ! [A: set_nat,B: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A,B)),A)) ).

% Diff_subset
tff(fact_1523_Diff__subset,axiom,
    ! [A: set_int,B: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A,B)),A)) ).

% Diff_subset
tff(fact_1524_Diff__mono,axiom,
    ! [A: set_nat,C: set_nat,D: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(D),B))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A,B)),minus_minus_set_nat(C,D))) ) ) ).

% Diff_mono
tff(fact_1525_Diff__mono,axiom,
    ! [A: set_int,C: set_int,D: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),C))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A,B)),minus_minus_set_int(C,D))) ) ) ).

% Diff_mono
tff(fact_1526_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_1527_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1528_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_1529_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_1530_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_1531_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_1532_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% add1_zle_eq
tff(fact_1533_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_1534_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_1535_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
           => ( pp(aa_int_bool(P,I2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_1536_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% member_bound_height
tff(fact_1537_Tb_H__cnt,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))))) ).

% Tb'_cnt
tff(fact_1538_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_1539_t__build__cnt,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ).

% t_build_cnt
tff(fact_1540_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ) ).

% TBOUND_buildupi
tff(fact_1541_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% delete_bound_size_univ
tff(fact_1542_buildup__build__time,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))) ).

% buildup_build_time
tff(fact_1543_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Deg) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),Deg))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% height_double_log_univ_size
tff(fact_1544_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% delete_bound_size_univ'
tff(fact_1545_vebt__minti_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ! [A3: bool,B3: bool] : X2 != vEBT_Leafi(A3,B3)
     => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : X2 != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : X2 != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% vebt_minti.cases
tff(fact_1546_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_0_not
tff(fact_1547_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_1548_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_1549_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% T_vebt_buildupi_gq_0
tff(fact_1550_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)),N)) ) ).

% buildup_gives_valid
tff(fact_1551_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14) = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_1552_double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1553_double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1554_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1555_mult__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1556_mult__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1557_mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),C2) = aa_rat_rat(times_times_rat(B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1558_mult__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1559_mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1560_mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1561_mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1562_mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A2) = aa_rat_rat(times_times_rat(C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1563_mult__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1564_mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1565_mult__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_1566_mult__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1567_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1568_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1569_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1570_mult__zero__right,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1571_mult__zero__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1572_mult__zero__right,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1573_mult__zero__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_1574_mult__zero__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1575_mult__zero__left,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex),A2) = zero_zero_complex ).

% mult_zero_left
tff(fact_1576_mult__zero__left,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A2) = zero_zero_real ).

% mult_zero_left
tff(fact_1577_mult__zero__left,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A2) = zero_zero_rat ).

% mult_zero_left
tff(fact_1578_mult__zero__left,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mult_zero_left
tff(fact_1579_mult__zero__left,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A2) = zero_zero_int ).

% mult_zero_left
tff(fact_1580_bits__div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_1581_bits__div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1582_bits__div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_div_0
tff(fact_1583_bits__div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% bits_div_0
tff(fact_1584_div__by__0,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1585_div__by__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1586_div__by__0,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1587_div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1588_div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1589_div__0,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,zero_zero_complex),A2) = zero_zero_complex ).

% div_0
tff(fact_1590_div__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A2) = zero_zero_real ).

% div_0
tff(fact_1591_div__0,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A2) = zero_zero_rat ).

% div_0
tff(fact_1592_div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% div_0
tff(fact_1593_div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% div_0
tff(fact_1594_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_1595_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_1596_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1597_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_1598_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1599_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2)) ).

% bot_nat_0.extremum
tff(fact_1600_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_1601_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_1602_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1603_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1604_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1605_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1606_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1607_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1608_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_1609_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_1610_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),zero_zero_nat) = A2 ).

% max_nat.right_neutral
tff(fact_1611_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1612_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A2) = A2 ).

% max_nat.left_neutral
tff(fact_1613_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1614_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ).

% zero_comp_diff_simps(1)
tff(fact_1615_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ).

% zero_comp_diff_simps(1)
tff(fact_1616_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% zero_comp_diff_simps(1)
tff(fact_1617_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ).

% zero_comp_diff_simps(2)
tff(fact_1618_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ).

% zero_comp_diff_simps(2)
tff(fact_1619_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% zero_comp_diff_simps(2)
tff(fact_1620_sum__squares__eq__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3)) = zero_zero_real )
    <=> ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1621_sum__squares__eq__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y3),Y3)) = zero_zero_rat )
    <=> ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1622_sum__squares__eq__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3)) = zero_zero_int )
    <=> ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1623_mult__cancel__left1,axiom,
    ! [C2: complex,B2: complex] :
      ( ( C2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1624_mult__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1625_mult__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1626_mult__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1627_mult__cancel__left2,axiom,
    ! [C2: complex,A2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1628_mult__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1629_mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1630_mult__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1631_mult__cancel__right1,axiom,
    ! [C2: complex,B2: complex] :
      ( ( C2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1632_mult__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1633_mult__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1634_mult__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1635_mult__cancel__right2,axiom,
    ! [A2: complex,C2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_1636_mult__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1637_mult__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1638_mult__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1639_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_1640_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_1641_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1642_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_1643_div__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),A2) = one_one_complex ) ) ).

% div_self
tff(fact_1644_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ).

% div_self
tff(fact_1645_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A2),A2) = one_one_rat ) ) ).

% div_self
tff(fact_1646_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),A2) = one_one_nat ) ) ).

% div_self
tff(fact_1647_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),A2) = one_one_int ) ) ).

% div_self
tff(fact_1648_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1649_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1650_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1651_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1652_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1653_nonzero__mult__div__cancel__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1654_nonzero__mult__div__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1655_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1656_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1657_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1658_div__mult__mult1__if,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ( C2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) = zero_zero_nat ) )
      & ( ( C2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% div_mult_mult1_if
tff(fact_1659_div__mult__mult1__if,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ( C2 = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) = zero_zero_int ) )
      & ( ( C2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% div_mult_mult1_if
tff(fact_1660_div__mult__mult2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ).

% div_mult_mult2
tff(fact_1661_div__mult__mult2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ).

% div_mult_mult2
tff(fact_1662_div__mult__mult1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ).

% div_mult_mult1
tff(fact_1663_div__mult__mult1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ).

% div_mult_mult1
tff(fact_1664_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_1665_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_1666_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_1667_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_1668_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_1669_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,N)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_1670_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_1671_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_1672_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_1673_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1674_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_1675_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_1676_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1677_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1678_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1679_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1680_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_1681_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1682_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1683_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1684_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1685_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_1686_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% semiring_1_class.of_nat_0
tff(fact_1687_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_1688_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_1689_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_1690_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_1691_power__Suc0__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1692_power__Suc0__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1693_power__Suc0__right,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1694_power__Suc0__right,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1695_power__Suc0__right,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1696_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_1697_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1698_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_1699_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_1700_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_1701_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1702_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1703_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_1704_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_1705_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_1706_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1707_max__0__1_I4_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X2),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X2) ).

% max_0_1(4)
tff(fact_1708_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X2)),zero_zero_real) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(4)
tff(fact_1709_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X2)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(4)
tff(fact_1710_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X2)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(4)
tff(fact_1711_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X2)),zero_zero_int) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(4)
tff(fact_1712_max__0__1_I3_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(X2)) = numera1916890842035813515d_enat(X2) ).

% max_0_1(3)
tff(fact_1713_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X2)) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(3)
tff(fact_1714_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,X2)) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(3)
tff(fact_1715_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X2)) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(3)
tff(fact_1716_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X2)) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(3)
tff(fact_1717_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1718_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_1719_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_1720_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_1721_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_1722_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_1723_max__0__1_I1_J,axiom,
    ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_1724_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_1725_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_1726_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_1727_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_1728_max__0__1_I2_J,axiom,
    ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_1729_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_1730_nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1731_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_1732_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1733_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_1734_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: complex,B2: complex,W: num] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),numera6690914467698888265omplex(W)) )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(W)) = B2 ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1735_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W)) = B2 ) )
        & ( ( aa_num_real(numeral_numeral_real,W) = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1736_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2 = aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W)) = B2 ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1737_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W: num,A2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),numera6690914467698888265omplex(W)) = A2 )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(W)) ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1738_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)) = A2 )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W) != zero_zero_real )
         => ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W) = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1739_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)) = A2 )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
         => ( B2 = aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1740_div__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self4
tff(fact_1741_div__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)),A2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self4
tff(fact_1742_div__mult__self3,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self3
tff(fact_1743_div__mult__self3,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)),A2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self3
tff(fact_1744_div__mult__self2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self2
tff(fact_1745_div__mult__self2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self2
tff(fact_1746_div__mult__self1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self1
tff(fact_1747_div__mult__self1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self1
tff(fact_1748_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1749_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1750_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1751_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1752_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A2),N) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1753_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1754_power__eq__0__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1755_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1756_power__eq__0__iff,axiom,
    ! [A2: complex,N: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1757_power__eq__0__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = zero_z3403309356797280102nteger )
    <=> ( ( A2 = zero_z3403309356797280102nteger )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_1758_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_1759_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_1760_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_1761_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1762_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_1763_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1764_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_1765_zero__eq__power2,axiom,
    ! [A2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1766_zero__eq__power2,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A2 = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1767_zero__eq__power2,axiom,
    ! [A2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1768_zero__eq__power2,axiom,
    ! [A2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1769_zero__eq__power2,axiom,
    ! [A2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1770_zero__eq__power2,axiom,
    ! [A2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_1771_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1772_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1773_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B2),M)),aa_nat_rat(power_power_rat(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1774_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1775_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1776_power__mono__iff,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1777_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
          <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1778_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1779_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1780_power__mono__iff,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_1781_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1782_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1783_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1784_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_1785_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_1786_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_1787_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1788_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_1789_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1790_power2__eq__iff__nonneg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1791_power2__eq__iff__nonneg,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3))
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1792_power2__eq__iff__nonneg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => ( ( aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1793_power2__eq__iff__nonneg,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y3))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1794_power2__eq__iff__nonneg,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1795_power2__less__eq__zero__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A2 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1796_power2__less__eq__zero__iff,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger))
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_1797_power2__less__eq__zero__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat))
    <=> ( A2 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_1798_power2__less__eq__zero__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int))
    <=> ( A2 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1799_zero__less__power2,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_1800_zero__less__power2,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1801_zero__less__power2,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1802_zero__less__power2,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1803_sum__power2__eq__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1804_sum__power2__eq__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1805_sum__power2__eq__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1806_sum__power2__eq__zero__iff,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger )
    <=> ( ( X2 = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1807_power__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_1808_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_1809_power__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),M)),aa_nat_rat(power_power_rat(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_1810_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_1811_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_1812_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1813_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1814_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1815_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1816_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1817_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M5: nat,N3: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M5)),aa_nat_int(semiri1314217659103216013at_int,N3)) ).

% int_diff_cases
tff(fact_1818_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_1819_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_1820_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_1821_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_1822_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_1823_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
tff(fact_1824_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_1825_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_1826_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_1827_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_1828_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_1829_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_1830_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_1831_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_1832_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_1833_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_1834_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_1835_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_1836_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_1837_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_1838_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D22))
       => ? [E2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1839_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D22))
       => ? [E2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1840_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_1841_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_1842_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_1843_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_1844_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_1845_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1846_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1847_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1848_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1849_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1850_mult__right__cancel,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1851_mult__right__cancel,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1852_mult__right__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A2),C2) = aa_rat_rat(times_times_rat(B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1853_mult__right__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1854_mult__right__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1855_mult__left__cancel,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1856_mult__left__cancel,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1857_mult__left__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C2),A2) = aa_rat_rat(times_times_rat(C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1858_mult__left__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1859_mult__left__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1860_no__zero__divisors,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_1861_no__zero__divisors,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1862_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A2),B2) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1863_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_1864_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1865_divisors__zero,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2) = zero_zero_complex )
     => ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_1866_divisors__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) = zero_zero_real )
     => ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1867_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1868_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_1869_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1870_mult__not__zero,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2) != zero_zero_complex )
     => ( ( A2 != zero_zero_complex )
        & ( B2 != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_1871_mult__not__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) != zero_zero_real )
     => ( ( A2 != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1872_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1873_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_1874_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1875_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A2),N) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1876_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: nat,N: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1877_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( A2 != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1878_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1879_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: complex,N: nat] :
      ( ( A2 != zero_zero_complex )
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1880_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1881_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1882_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% not0_implies_Suc
tff(fact_1883_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1884_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1885_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1886_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P,N3)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_1887_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),zero_zero_nat))
     => ( ! [Y4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y4)))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_1888_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,N3))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_1889_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y3 != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1890_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1891_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1892_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1893_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_1894_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( ~ pp(aa_nat_bool(P,N3))
             => ? [M3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
                  & ~ pp(aa_nat_bool(P,M3)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_1895_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1896_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_1897_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_1898_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1899_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_1900_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_1901_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1902_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_1903_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_1904_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
    <=> ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1905_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1906_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1907_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1908_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_1909_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1910_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_1911_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1912_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1913_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
         => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1914_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
         => ( ( aa_nat_rat(power_power_rat(A2),N) = aa_nat_rat(power_power_rat(B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1915_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1916_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1917_power__eq__imp__eq__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1918_power__eq__imp__eq__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B2),N) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1919_power__eq__imp__eq__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),N) = aa_nat_rat(power_power_rat(B2),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1920_power__eq__imp__eq__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1921_power__eq__imp__eq__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1922_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_1923_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_1924_lambda__zero,axiom,
    aTP_Lamm_bg = aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex) ).

% lambda_zero
tff(fact_1925_lambda__zero,axiom,
    aTP_Lamm_bh = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_1926_lambda__zero,axiom,
    aTP_Lamm_bi = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_1927_lambda__zero,axiom,
    aTP_Lamm_bj = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_1928_lambda__zero,axiom,
    aTP_Lamm_bk = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_1929_power__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1930_power__strict__mono,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1931_power__strict__mono,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1932_power__strict__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1933_power__strict__mono,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N))) ) ) ) ).

% power_strict_mono
tff(fact_1934_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_1935_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).

% zadd_int_left
tff(fact_1936_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z))
    <=> ? [N7: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N7)) ) ).

% zle_iff_zadd
tff(fact_1937_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% zdiv_int
tff(fact_1938_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_le_numeral
tff(fact_1939_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_le_numeral
tff(fact_1940_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_1941_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_le_numeral
tff(fact_1942_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_1943_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_1944_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_1945_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_1946_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_1947_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_1948_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_1949_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_1950_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_less_numeral
tff(fact_1951_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_less_numeral
tff(fact_1952_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_1953_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_less_numeral
tff(fact_1954_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_1955_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_1956_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_1957_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_1958_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1959_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1960_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1961_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1962_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_1963_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1964_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1965_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_1966_mult__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_1967_mult__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_1968_mult__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_1969_mult__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_1970_mult__mono_H,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_1971_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_1972_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_1973_mult__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_1974_zero__le__square,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2))) ).

% zero_le_square
tff(fact_1975_zero__le__square,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),A2))) ).

% zero_le_square
tff(fact_1976_zero__le__square,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2))) ).

% zero_le_square
tff(fact_1977_split__mult__pos__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))) ) ).

% split_mult_pos_le
tff(fact_1978_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))) ) ).

% split_mult_pos_le
tff(fact_1979_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ).

% split_mult_pos_le
tff(fact_1980_mult__left__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_1981_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_1982_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_1983_mult__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_1984_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_1985_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_1986_mult__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))) ) ) ).

% mult_left_mono
tff(fact_1987_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))) ) ) ).

% mult_left_mono
tff(fact_1988_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))) ) ) ).

% mult_left_mono
tff(fact_1989_mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ).

% mult_left_mono
tff(fact_1990_mult__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_1991_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_1992_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_1993_mult__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ) ) ).

% mult_right_mono
tff(fact_1994_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))) ) ) ).

% mult_right_mono
tff(fact_1995_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))) ) ) ).

% mult_right_mono
tff(fact_1996_mult__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))) ) ) ).

% mult_right_mono
tff(fact_1997_mult__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_1998_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_1999_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_2000_split__mult__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_2001_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_2002_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_2003_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_2004_mult__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2005_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2006_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2007_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2008_mult__nonneg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2009_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2010_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2011_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2012_mult__nonpos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2013_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2014_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2015_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2016_mult__nonneg__nonpos2,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2017_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(B2),A2)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2018_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2019_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2020_zero__le__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2021_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2022_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2023_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2024_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2025_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2026_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2027_add__less__zeroD,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_2028_add__less__zeroD,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(X2),Y3)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_2029_add__less__zeroD,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_2030_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_2031_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_2032_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_2033_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_2034_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_2035_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_2036_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_2037_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_2038_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_2039_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_2040_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_2041_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_2042_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2043_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2044_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2045_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2046_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2047_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2048_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2049_mult__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2050_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2051_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2052_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2053_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2054_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2055_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2056_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2057_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2058_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2059_mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2060_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2061_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2062_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_2063_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2064_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2065_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2066_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2067_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2068_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2069_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2070_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2071_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2072_zero__less__mult__pos2,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2073_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B2),A2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2074_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2075_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_2076_zero__less__mult__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2077_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2078_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2079_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_2080_zero__less__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2081_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2082_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2083_mult__pos__neg2,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_2084_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(B2),A2)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_2085_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_2086_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_2087_mult__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_2088_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_2089_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_2090_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_2091_mult__pos__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_2092_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_2093_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_2094_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_2095_mult__neg__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_2096_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_2097_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_2098_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_2099_mult__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_2100_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_2101_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_2102_not__square__less__zero,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_2103_not__square__less__zero,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),A2)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_2104_not__square__less__zero,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_2105_mult__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))) ) ) ).

% mult_neg_neg
tff(fact_2106_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))) ) ) ).

% mult_neg_neg
tff(fact_2107_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_neg_neg
tff(fact_2108_power__mono,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))) ) ) ).

% power_mono
tff(fact_2109_power__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))) ) ) ).

% power_mono
tff(fact_2110_power__mono,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N))) ) ) ).

% power_mono
tff(fact_2111_power__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))) ) ) ).

% power_mono
tff(fact_2112_power__mono,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N))) ) ) ).

% power_mono
tff(fact_2113_zero__le__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% zero_le_power
tff(fact_2114_zero__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% zero_le_power
tff(fact_2115_zero__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% zero_le_power
tff(fact_2116_zero__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% zero_le_power
tff(fact_2117_zero__le__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ).

% zero_le_power
tff(fact_2118_zero__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% zero_less_power
tff(fact_2119_zero__less__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% zero_less_power
tff(fact_2120_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% zero_less_power
tff(fact_2121_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% zero_less_power
tff(fact_2122_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ).

% zero_less_power
tff(fact_2123_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_2124_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_2125_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_2126_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_2127_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_2128_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_2129_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_2130_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_2131_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2132_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2133_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2134_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2135_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2136_power__0,axiom,
    ! [A2: assn] : aa_nat_assn(power_power_assn(A2),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_2137_power__0,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2138_power__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2139_power__0,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2140_power__0,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2141_power__0,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2142_power__0,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_2143_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2144_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% gr0_implies_Suc
tff(fact_2145_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_2146_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M7: nat] : N = aa_nat_nat(suc,M7) ) ).

% gr0_conv_Suc
tff(fact_2147_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_2148_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_2149_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_2150_nat__compl__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N3))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct
tff(fact_2151_nat__compl__induct_H,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N3))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct'
tff(fact_2152_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2153_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(some_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2154_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2155_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2156_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2157_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2158_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
            & ! [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K2))
               => ~ pp(aa_nat_bool(P,I5)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_2159_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(suc,N) = M )
     => ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_2160_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2161_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_2162_nat__geq__1__eq__neqz,axiom,
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),X2))
    <=> ( X2 != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_2163_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2164_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_2165_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_2166_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_2167_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_2168_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_2169_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2170_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S2)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S2) ).

% vebt_insert.simps(2)
tff(fact_2171_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
    <=> ? [N7: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N7))) ) ).

% zless_iff_Suc_zadd
tff(fact_2172_mult__less__le__imp__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2173_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2174_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2175_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2176_mult__le__less__imp__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2177_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2178_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2179_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2180_mult__right__le__imp__le,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2181_mult__right__le__imp__le,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2182_mult__right__le__imp__le,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2183_mult__right__le__imp__le,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_2184_mult__left__le__imp__le,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2185_mult__left__le__imp__le,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2186_mult__left__le__imp__le,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2187_mult__left__le__imp__le,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_2188_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2189_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2190_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2191_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2192_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2193_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2194_mult__less__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2195_mult__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2196_mult__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2197_mult__strict__mono_H,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2198_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2199_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2200_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2201_mult__right__less__imp__less,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2202_mult__right__less__imp__less,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2203_mult__right__less__imp__less,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2204_mult__right__less__imp__less,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_2205_mult__less__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2206_mult__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2207_mult__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2208_mult__strict__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2209_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2210_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2211_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2212_mult__left__less__imp__less,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2213_mult__left__less__imp__less,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2214_mult__left__less__imp__less,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2215_mult__left__less__imp__less,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_2216_mult__le__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2217_mult__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2218_mult__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2219_mult__le__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2220_mult__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2221_mult__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2222_sum__squares__ge__zero,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3)))) ).

% sum_squares_ge_zero
tff(fact_2223_sum__squares__ge__zero,axiom,
    ! [X2: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y3),Y3)))) ).

% sum_squares_ge_zero
tff(fact_2224_sum__squares__ge__zero,axiom,
    ! [X2: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3)))) ).

% sum_squares_ge_zero
tff(fact_2225_sum__squares__le__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3))),zero_zero_real))
    <=> ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2226_sum__squares__le__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y3),Y3))),zero_zero_rat))
    <=> ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2227_sum__squares__le__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3))),zero_zero_int))
    <=> ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2228_mult__left__le__one__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),X2)),X2)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2229_mult__left__le__one__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y3),X2)),X2)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2230_mult__left__le__one__le,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y3),X2)),X2)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2231_mult__right__le__one__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)),X2)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2232_mult__right__le__one__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X2),Y3)),X2)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2233_mult__right__le__one__le,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y3)),X2)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2234_mult__le__one,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_2235_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_2236_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_2237_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_2238_mult__left__le,axiom,
    ! [C2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),A2)) ) ) ).

% mult_left_le
tff(fact_2239_mult__left__le,axiom,
    ! [C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),A2)) ) ) ).

% mult_left_le
tff(fact_2240_mult__left__le,axiom,
    ! [C2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),A2)) ) ) ).

% mult_left_le
tff(fact_2241_mult__left__le,axiom,
    ! [C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),A2)) ) ) ).

% mult_left_le
tff(fact_2242_div__positive,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) ) ) ).

% div_positive
tff(fact_2243_div__positive,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) ) ) ).

% div_positive
tff(fact_2244_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2245_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2246_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_2247_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_2248_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_2249_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_2250_not__sum__squares__lt__zero,axiom,
    ! [X2: real,Y3: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_2251_not__sum__squares__lt__zero,axiom,
    ! [X2: rat,Y3: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y3),Y3))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_2252_not__sum__squares__lt__zero,axiom,
    ! [X2: int,Y3: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2253_sum__squares__gt__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3))))
    <=> ( ( X2 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2254_sum__squares__gt__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y3),Y3))))
    <=> ( ( X2 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2255_sum__squares__gt__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Y3))))
    <=> ( ( X2 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2256_power__less__imp__less__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_2257_power__less__imp__less__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_2258_power__less__imp__less__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_2259_power__less__imp__less__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_2260_power__less__imp__less__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% power_less_imp_less_base
tff(fact_2261_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2262_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2263_power__le__one,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2264_power__le__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),one_one_Code_integer)) ) ) ).

% power_le_one
tff(fact_2265_power__le__one,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_2266_power__le__one,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2267_power__le__one,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2268_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: complex,C2: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),C2) = B2 ) )
        & ( ( C2 = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2269_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2) = B2 ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2270_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2) = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2271_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C2: complex,W: num] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) = numera6690914467698888265omplex(W) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2272_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = aa_num_real(numeral_numeral_real,W) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2273_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C2) = aa_num_rat(numeral_numeral_rat,W) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B2 = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2274_power__inject__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2275_power__inject__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2276_power__inject__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2277_power__inject__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2278_power__inject__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2279_power__le__imp__le__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_2280_power__le__imp__le__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(B2),aa_nat_nat(suc,N))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_2281_power__le__imp__le__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_2282_power__le__imp__le__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_2283_power__le__imp__le__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% power_le_imp_le_base
tff(fact_2284_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_2285_div__add__self2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2286_div__add__self2,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2287_div__add__self1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2288_div__add__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2289_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_2290_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_2291_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
            & ! [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),K2))
               => ~ pp(aa_nat_bool(P,I5)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_2292_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(P,N3))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_2293_num_Osize_I6_J,axiom,
    ! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_2294_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_2295_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_2296_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_2297_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% one_less_mult
tff(fact_2298_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_2299_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_2300_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_2301_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D3: nat] :
            ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
           => pp(aa_nat_bool(P,D3)) ) ) ) ).

% nat_diff_split
tff(fact_2302_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D3: nat] :
              ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
              & ~ pp(aa_nat_bool(P,D3)) ) ) ) ).

% nat_diff_split_asm
tff(fact_2303_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_2304_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

% nat_one_le_power
tff(fact_2305_length__pos__if__in__set,axiom,
    ! [X2: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2306_length__pos__if__in__set,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2307_length__pos__if__in__set,axiom,
    ! [X2: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2308_length__pos__if__in__set,axiom,
    ! [X2: bool,Xs: list_bool] :
      ( member_bool(X2,set_bool2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2309_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2310_length__pos__if__in__set,axiom,
    ! [X2: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2311_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_2312_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_2313_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3))) ) ) ).

% div_less_iff_less_mult
tff(fact_2314_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),B2)) ) ) ).

% td_gal_lt
tff(fact_2315_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S2)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S2) ).

% vebt_insert.simps(3)
tff(fact_2316_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),X2)) ).

% vebt_member.simps(3)
tff(fact_2317_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S2)),X2) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_2318_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_2319_mult__less__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2320_mult__less__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2321_mult__less__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2322_mult__less__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2323_mult__less__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(times_times_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2324_mult__less__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2325_mult__less__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2326_mult__less__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2327_mult__less__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2328_mult__less__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2329_mult__less__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(times_times_rat(C2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2330_mult__less__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2331_mult__le__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2332_mult__le__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2333_mult__le__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2334_mult__le__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2335_mult__le__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2336_mult__le__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2337_mult__le__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2338_mult__le__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2339_mult__le__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2340_mult__le__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2341_mult__le__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(C2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2342_mult__le__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2343_convex__bound__le,axiom,
    ! [X2: real,A2: real,Y3: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y3))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2344_convex__bound__le,axiom,
    ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X2)),aa_rat_rat(times_times_rat(V),Y3))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2345_convex__bound__le,axiom,
    ! [X2: int,A2: int,Y3: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y3))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2346_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2347_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2348_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_num_real(numeral_numeral_real,W)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2349_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_num_rat(numeral_numeral_rat,W)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2350_power__Suc__less,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ) ).

% power_Suc_less
tff(fact_2351_power__Suc__less,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ).

% power_Suc_less
tff(fact_2352_power__Suc__less,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N))),aa_nat_rat(power_power_rat(A2),N))) ) ) ).

% power_Suc_less
tff(fact_2353_power__Suc__less,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ).

% power_Suc_less
tff(fact_2354_power__Suc__less,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ) ).

% power_Suc_less
tff(fact_2355_power__Suc__le__self,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_2356_power__Suc__le__self,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_2357_power__Suc__le__self,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_2358_power__Suc__le__self,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_2359_power__Suc__le__self,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N))),A2)) ) ) ).

% power_Suc_le_self
tff(fact_2360_power__Suc__less__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))),one_one_Code_integer)) ) ) ).

% power_Suc_less_one
tff(fact_2361_power__Suc__less__one,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_2362_power__Suc__less__one,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_2363_power__Suc__less__one,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_2364_power__Suc__less__one,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_2365_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_2366_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_2367_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_2368_zero__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_2369_zero__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_2370_zero__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_2371_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N6)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2372_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N6)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2373_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N6)),aa_nat_rat(power_power_rat(A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2374_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N6)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2375_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N6)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2376_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N6)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ) ).

% power_decreasing
tff(fact_2377_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N6)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ) ) ).

% power_decreasing
tff(fact_2378_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N6)),aa_nat_rat(power_power_rat(A2),N))) ) ) ) ).

% power_decreasing
tff(fact_2379_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N6)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ) ).

% power_decreasing
tff(fact_2380_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N6)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ) ) ).

% power_decreasing
tff(fact_2381_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_2382_self__le__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ).

% self_le_power
tff(fact_2383_self__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ) ).

% self_le_power
tff(fact_2384_self__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(power_power_rat(A2),N))) ) ) ).

% self_le_power
tff(fact_2385_self__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ).

% self_le_power
tff(fact_2386_self__le__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ) ).

% self_le_power
tff(fact_2387_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_2388_power__diff,axiom,
    ! [A2: code_integer,N: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ) ).

% power_diff
tff(fact_2389_power__diff,axiom,
    ! [A2: complex,N: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ) ) ) ).

% power_diff
tff(fact_2390_power__diff,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ) ).

% power_diff
tff(fact_2391_power__diff,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)) ) ) ) ).

% power_diff
tff(fact_2392_power__diff,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_diff
tff(fact_2393_power__diff,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ) ) ) ).

% power_diff
tff(fact_2394_one__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ) ).

% one_less_power
tff(fact_2395_one__less__power,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ) ).

% one_less_power
tff(fact_2396_one__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N))) ) ) ).

% one_less_power
tff(fact_2397_one__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ) ).

% one_less_power
tff(fact_2398_one__less__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ) ).

% one_less_power
tff(fact_2399_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_2400_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M)) ) ) ).

% nz_le_conv_less
tff(fact_2401_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2402_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_2403_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_if
tff(fact_2404_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_2405_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_2406_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),K) ) ) ).

% msrevs(1)
tff(fact_2407_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).

% dividend_less_times_div
tff(fact_2408_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_2409_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_2410_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q3)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2411_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2))) ) ) ).

% td_gal
tff(fact_2412_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_2413_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))) ) ) ).

% nat_mult_power_less_eq
tff(fact_2414_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X2: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry) ).

% vebt_delete.simps(5)
tff(fact_2415_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X2)) ).

% vebt_member.simps(4)
tff(fact_2416_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S2)),X2) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_2417_convex__bound__lt,axiom,
    ! [X2: real,A2: real,Y3: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y3))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2418_convex__bound__lt,axiom,
    ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X2)),aa_rat_rat(times_times_rat(V),Y3))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2419_convex__bound__lt,axiom,
    ! [X2: int,A2: int,Y3: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y3))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2420_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2421_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2422_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_num_real(numeral_numeral_real,W)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2423_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_num_rat(numeral_numeral_rat,W)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2424_half__gt__zero__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% half_gt_zero_iff
tff(fact_2425_half__gt__zero__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% half_gt_zero_iff
tff(fact_2426_half__gt__zero,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2427_half__gt__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2428_power2__le__imp__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ).

% power2_le_imp_le
tff(fact_2429_power2__le__imp__le,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X2),Y3)) ) ) ).

% power2_le_imp_le
tff(fact_2430_power2__le__imp__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ) ).

% power2_le_imp_le
tff(fact_2431_power2__le__imp__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ).

% power2_le_imp_le
tff(fact_2432_power2__le__imp__le,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ) ).

% power2_le_imp_le
tff(fact_2433_power2__eq__imp__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2434_power2__eq__imp__eq,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3))
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2435_power2__eq__imp__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2436_power2__eq__imp__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y3))
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2437_power2__eq__imp__eq,axiom,
    ! [X2: int,Y3: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2438_zero__le__power2,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2439_zero__le__power2,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2440_zero__le__power2,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2441_zero__le__power2,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2442_power2__less__0,axiom,
    ! [A2: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger)) ).

% power2_less_0
tff(fact_2443_power2__less__0,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_2444_power2__less__0,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_2445_power2__less__0,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_2446_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2447_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2448_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2449_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2450_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2451_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2452_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2453_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2454_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2455_power__diff__power__eq,axiom,
    ! [A2: code_integer,N: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_2456_power__diff__power__eq,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_2457_power__diff__power__eq,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_2458_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_2459_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_2460_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_2461_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_2462_power__eq__if,axiom,
    ! [M: nat,P2: assn] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_assn(power_power_assn(P2),M) = one_one_assn ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_assn(power_power_assn(P2),M) = aa_assn_assn(times_times_assn(P2),aa_nat_assn(power_power_assn(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2463_power__eq__if,axiom,
    ! [M: nat,P2: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),M) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,P2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2464_power__eq__if,axiom,
    ! [M: nat,P2: code_integer] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(P2),M) = one_one_Code_integer ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(P2),M) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(P2),aa_nat_Code_integer(power_8256067586552552935nteger(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2465_power__eq__if,axiom,
    ! [M: nat,P2: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2466_power__eq__if,axiom,
    ! [M: nat,P2: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P2),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P2),M) = aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2467_power__eq__if,axiom,
    ! [M: nat,P2: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2468_power__eq__if,axiom,
    ! [M: nat,P2: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2469_power__minus__mult,axiom,
    ! [N: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) ) ) ).

% power_minus_mult
tff(fact_2470_power__minus__mult,axiom,
    ! [N: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) ) ) ).

% power_minus_mult
tff(fact_2471_power__minus__mult,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) ) ) ).

% power_minus_mult
tff(fact_2472_power__minus__mult,axiom,
    ! [N: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_rat(power_power_rat(A2),N) ) ) ).

% power_minus_mult
tff(fact_2473_power__minus__mult,axiom,
    ! [N: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) ) ) ).

% power_minus_mult
tff(fact_2474_power__minus__mult,axiom,
    ! [N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) ) ) ).

% power_minus_mult
tff(fact_2475_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_2476_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q5)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q5))))
            & pp(aa_nat_bool(P,Q5)) ) ) ) ).

% split_div'
tff(fact_2477_power__sub,axiom,
    ! [N: nat,M: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_sub
tff(fact_2478_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X2: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm)),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) ).

% vebt_delete.simps(6)
tff(fact_2479_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),X2) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_2480_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),X2) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_2481_power2__less__imp__less,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y3))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y3)) ) ) ).

% power2_less_imp_less
tff(fact_2482_power2__less__imp__less,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ).

% power2_less_imp_less
tff(fact_2483_power2__less__imp__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ) ).

% power2_less_imp_less
tff(fact_2484_power2__less__imp__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ) ).

% power2_less_imp_less
tff(fact_2485_power2__less__imp__less,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3)) ) ) ).

% power2_less_imp_less
tff(fact_2486_sum__power2__le__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2487_sum__power2__le__zero__iff,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger))
    <=> ( ( X2 = zero_z3403309356797280102nteger )
        & ( Y3 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2488_sum__power2__le__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2489_sum__power2__le__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
    <=> ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2490_sum__power2__ge__zero,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2491_sum__power2__ge__zero,axiom,
    ! [X2: code_integer,Y3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2492_sum__power2__ge__zero,axiom,
    ! [X2: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2493_sum__power2__ge__zero,axiom,
    ! [X2: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2494_sum__power2__gt__zero__iff,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X2 != zero_z3403309356797280102nteger )
        | ( Y3 != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2495_sum__power2__gt__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X2 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2496_sum__power2__gt__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X2 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2497_sum__power2__gt__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X2 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2498_not__sum__power2__lt__zero,axiom,
    ! [X2: code_integer,Y3: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)) ).

% not_sum_power2_lt_zero
tff(fact_2499_not__sum__power2__lt__zero,axiom,
    ! [X2: real,Y3: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_2500_not__sum__power2__lt__zero,axiom,
    ! [X2: rat,Y3: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_2501_not__sum__power2__lt__zero,axiom,
    ! [X2: int,Y3: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_2502_zero__le__even__power_H,axiom,
    ! [A2: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_2503_zero__le__even__power_H,axiom,
    ! [A2: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_2504_zero__le__even__power_H,axiom,
    ! [A2: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_2505_zero__le__even__power_H,axiom,
    ! [A2: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_2506_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2507_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_2508_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))) ) )
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P,N3))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_2509_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),X2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_2510_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X2) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_2511_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X2) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_2512_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2513_odd__0__le__power__imp__0__le,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2514_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2515_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2516_odd__power__less__zero,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_z3403309356797280102nteger)) ) ).

% odd_power_less_zero
tff(fact_2517_odd__power__less__zero,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_2518_odd__power__less__zero,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_2519_odd__power__less__zero,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_2520_VEBTi_Oexhaust,axiom,
    ! [Y3: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] : Y3 != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X112,X122,X132),X142)
     => ~ ! [X212: bool,X223: bool] : Y3 != vEBT_Leafi(X212,X223) ) ).

% VEBTi.exhaust
tff(fact_2521_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: bool,X222: bool] : aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14) != vEBT_Leafi(X21,X222) ).

% VEBTi.distinct(1)
tff(fact_2522_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2523_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2524_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_2525_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_2526_arith__geo__mean,axiom,
    ! [U: real,X2: real,Y3: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_2527_arith__geo__mean,axiom,
    ! [U: rat,X2: rat,Y3: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X2),Y3) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),Y3)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_2528_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% member_bound_size_univ
tff(fact_2529_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ( X2 != vEBT_Leafi(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X2 != vEBT_Leafi(fTrue,Uv2)
       => ( ! [Uu2: bool] : X2 != vEBT_Leafi(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : X2 != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : X2 != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_2530_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% Tb_T_vebt_buildupi
tff(fact_2531_le__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2532_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2533_le__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2534_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2535_divide__le__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2536_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2537_divide__le__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2538_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2539_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_2540_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_2541_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_TBOUND_bool(vEBT_VEBT_minNulli(T)),one_one_nat)) ).

% TBOUND_minNulli
tff(fact_2542_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2543_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2544_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(one_one_rat),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2545_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2546_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2547_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(one_one_rat),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2548_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T))) ).

% cnt_non_neg
tff(fact_2549_Tb__Tb_H,axiom,
    ! [T: nat] : aa_nat_int(vEBT_VEBT_Tb,T) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,T)) ).

% Tb_Tb'
tff(fact_2550_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_2551_idiff__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_2552_idiff__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_2553_divide__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2554_divide__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2555_divide__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2556_divide__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,C2),A2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_2557_divide__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_2558_divide__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C2),A2) = aa_rat_rat(divide_divide_rat(C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_2559_divide__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_2560_divide__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_2561_divide__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),C2) = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_2562_division__ring__divide__zero,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2563_division__ring__divide__zero,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2564_division__ring__divide__zero,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2565_times__divide__eq__left,axiom,
    ! [B2: complex,C2: complex,A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2)),A2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),A2)),C2) ).

% times_divide_eq_left
tff(fact_2566_times__divide__eq__left,axiom,
    ! [B2: real,C2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),C2) ).

% times_divide_eq_left
tff(fact_2567_times__divide__eq__left,axiom,
    ! [B2: rat,C2: rat,A2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B2),C2)),A2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B2),A2)),C2) ).

% times_divide_eq_left
tff(fact_2568_divide__divide__eq__left,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)),C2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2)) ).

% divide_divide_eq_left
tff(fact_2569_divide__divide__eq__left,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% divide_divide_eq_left
tff(fact_2570_divide__divide__eq__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A2),B2)),C2) = aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(times_times_rat(B2),C2)) ).

% divide_divide_eq_left
tff(fact_2571_divide__divide__eq__right,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2)),B2) ).

% divide_divide_eq_right
tff(fact_2572_divide__divide__eq__right,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2) ).

% divide_divide_eq_right
tff(fact_2573_divide__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(divide_divide_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),C2)),B2) ).

% divide_divide_eq_right
tff(fact_2574_times__divide__eq__right,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)),C2) ).

% times_divide_eq_right
tff(fact_2575_times__divide__eq__right,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2) ).

% times_divide_eq_right
tff(fact_2576_times__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(divide_divide_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2)),C2) ).

% times_divide_eq_right
tff(fact_2577_log__one,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A2),one_one_real) = zero_zero_real ).

% log_one
tff(fact_2578_divide__eq__1__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) = one_one_complex )
    <=> ( ( B2 != zero_zero_complex )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2579_divide__eq__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = one_one_real )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2580_divide__eq__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),B2) = one_one_rat )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2581_one__eq__divide__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( one_one_complex = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2582_one__eq__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2583_one__eq__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A2),B2) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2584_divide__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),A2) = one_one_complex ) ) ).

% divide_self
tff(fact_2585_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ).

% divide_self
tff(fact_2586_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A2),A2) = one_one_rat ) ) ).

% divide_self
tff(fact_2587_divide__self__if,axiom,
    ! [A2: complex] :
      ( ( ( A2 = zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),A2) = zero_zero_complex ) )
      & ( ( A2 != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),A2) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_2588_divide__self__if,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_2589_divide__self__if,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A2),A2) = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A2),A2) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_2590_divide__eq__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2) = one_one_real )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_2591_divide__eq__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),A2) = one_one_rat )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_2592_eq__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2) )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_2593_eq__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B2),A2) )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_2594_one__divide__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2595_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2596_zero__eq__1__divide__iff,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2597_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2598_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( ( C2 = zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2)) = zero_zero_complex ) )
      & ( ( C2 != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2599_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ( C2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = zero_zero_real ) )
      & ( ( C2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2600_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ( C2 = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) = zero_zero_rat ) )
      & ( ( C2 != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2601_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2602_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2603_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2604_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2605_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2606_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2607_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2608_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2609_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2610_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2611_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2612_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(C2),B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2613_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2614_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2615_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = A2 )
      <=> ( B2 = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_2616_log__eq__one,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),A2) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_2617_log__less__cancel__iff,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y3)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2618_log__less__one__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),A2)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2619_one__less__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X2)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2620_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_2621_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2622_zero__le__divide__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% zero_le_divide_1_iff
tff(fact_2623_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% zero_le_divide_1_iff
tff(fact_2624_divide__le__0__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_2625_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_2626_divide__less__0__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_2627_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_2628_divide__less__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2629_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2630_divide__less__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2631_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2632_less__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2633_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2634_less__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2635_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2636_zero__less__divide__1__iff,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% zero_less_divide_1_iff
tff(fact_2637_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% zero_less_divide_1_iff
tff(fact_2638_log__le__cancel__iff,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y3)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2639_log__le__one__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),A2)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2640_one__le__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X2)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2641_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_2642_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2643_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_2644_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_2645_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),B2)) = aa_nat_real(semiri5074537144036343181t_real,B2) ) ) ) ).

% log_pow_cancel
tff(fact_2646_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_2647_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N3: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% zero_le_imp_eq_int
tff(fact_2648_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N3: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% nonneg_int_cases
tff(fact_2649_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_2650_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_2651_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_2652_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_2653_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,N)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_2654_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_2655_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),zero_z5237406670263579293d_enat))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_2656_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N)) ).

% i0_lb
tff(fact_2657_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_2658_linordered__field__no__lb,axiom,
    ! [X6: real] :
    ? [Y4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),X6)) ).

% linordered_field_no_lb
tff(fact_2659_linordered__field__no__lb,axiom,
    ! [X6: rat] :
    ? [Y4: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y4),X6)) ).

% linordered_field_no_lb
tff(fact_2660_linordered__field__no__ub,axiom,
    ! [X6: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),X_12)) ).

% linordered_field_no_ub
tff(fact_2661_linordered__field__no__ub,axiom,
    ! [X6: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),X_12)) ).

% linordered_field_no_ub
tff(fact_2662_log__base__change,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2663_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% pos_int_cases
tff(fact_2664_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2665_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_2666_real__arch__pow__inv,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N3)),Y3)) ) ) ).

% real_arch_pow_inv
tff(fact_2667_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_2668_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_2669_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2670_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_2671_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_2672_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_2673_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2674_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_2675_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_2676_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).

% div_positive_int
tff(fact_2677_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_2678_zdiv__mono2__neg,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B4)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2679_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_2680_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2681_zdiv__mono2,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B4))) ) ) ) ).

% zdiv_mono2
tff(fact_2682_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B2))) ) ) ).

% zdiv_mono1
tff(fact_2683_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),A2)) ) ) ).

% zdiv_le_dividend
tff(fact_2684_int__div__less__self,axiom,
    ! [X2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),K)),X2)) ) ) ).

% int_div_less_self
tff(fact_2685_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_2686_zdiv__mult__self,axiom,
    ! [M: int,A2: int,N: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))),M) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),M)),N) ) ) ).

% zdiv_mult_self
tff(fact_2687_log__mult,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y3)) ) ) ) ) ) ).

% log_mult
tff(fact_2688_log__divide,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Y3)) ) ) ) ) ) ).

% log_divide
tff(fact_2689_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_2690_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),N) = A2 )
            & ! [Y5: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y5))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y5),N) = A2 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2691_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),N) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_2692_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).

% le_imp_0_less
tff(fact_2693_q__pos__lemma,axiom,
    ! [B4: int,Q6: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_2694_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R2: int,B4: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B4))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q3),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2695_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R2: int,B4: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q3)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2696_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q3)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2697_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q3),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2698_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_2699_real__archimedian__rdiv__eq__0,axiom,
    ! [X2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M5)),X2)),C2)) )
         => ( X2 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2700_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_2701_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),R2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2702_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2703_real__of__nat__div2,axiom,
    ! [N: nat,X2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X2))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X2))))) ).

% real_of_nat_div2
tff(fact_2704_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K),M)),K) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_2705_axxdiv2,axiom,
    ! [X2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),X2)),X2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = X2 )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),X2)),X2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = X2 ) ) ).

% axxdiv2
tff(fact_2706_add__divide__distrib,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)),C2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2)) ).

% add_divide_distrib
tff(fact_2707_add__divide__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ).

% add_divide_distrib
tff(fact_2708_add__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ).

% add_divide_distrib
tff(fact_2709_times__divide__times__eq,axiom,
    ! [X2: complex,Y3: complex,Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),W)) ).

% times_divide_times_eq
tff(fact_2710_times__divide__times__eq,axiom,
    ! [X2: real,Y3: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),W)) ).

% times_divide_times_eq
tff(fact_2711_times__divide__times__eq,axiom,
    ! [X2: rat,Y3: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(X2),Y3)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(Y3),W)) ).

% times_divide_times_eq
tff(fact_2712_divide__divide__times__eq,axiom,
    ! [X2: complex,Y3: complex,Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),W)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z)) ).

% divide_divide_times_eq
tff(fact_2713_divide__divide__times__eq,axiom,
    ! [X2: real,Y3: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z)) ).

% divide_divide_times_eq
tff(fact_2714_divide__divide__times__eq,axiom,
    ! [X2: rat,Y3: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(X2),Y3)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X2),W)),aa_rat_rat(times_times_rat(Y3),Z)) ).

% divide_divide_times_eq
tff(fact_2715_divide__divide__eq__left_H,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)),C2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2)) ).

% divide_divide_eq_left'
tff(fact_2716_divide__divide__eq__left_H,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ).

% divide_divide_eq_left'
tff(fact_2717_divide__divide__eq__left_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A2),B2)),C2) = aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(times_times_rat(C2),B2)) ).

% divide_divide_eq_left'
tff(fact_2718_diff__divide__distrib,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2)),C2) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2)) ).

% diff_divide_distrib
tff(fact_2719_diff__divide__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ).

% diff_divide_distrib
tff(fact_2720_diff__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ).

% diff_divide_distrib
tff(fact_2721_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_2722_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_2723_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_2724_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_2725_divide__right__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2))) ) ) ).

% divide_right_mono_neg
tff(fact_2726_divide__right__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_rat_rat(divide_divide_rat(A2),C2))) ) ) ).

% divide_right_mono_neg
tff(fact_2727_divide__nonpos__nonpos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2728_divide__nonpos__nonpos,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2729_divide__nonpos__nonneg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2730_divide__nonpos__nonneg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2731_divide__nonneg__nonpos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2732_divide__nonneg__nonpos,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2733_divide__nonneg__nonneg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2734_divide__nonneg__nonneg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2735_zero__le__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2736_zero__le__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2737_divide__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))) ) ) ).

% divide_right_mono
tff(fact_2738_divide__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2))) ) ) ).

% divide_right_mono
tff(fact_2739_divide__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ) ) ).

% divide_le_0_iff
tff(fact_2740_divide__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),B2)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) ) ) ) ).

% divide_le_0_iff
tff(fact_2741_divide__neg__neg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% divide_neg_neg
tff(fact_2742_divide__neg__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% divide_neg_neg
tff(fact_2743_divide__neg__pos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_2744_divide__neg__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2745_divide__pos__neg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_2746_divide__pos__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2747_divide__pos__pos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% divide_pos_pos
tff(fact_2748_divide__pos__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% divide_pos_pos
tff(fact_2749_divide__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ) ) ).

% divide_less_0_iff
tff(fact_2750_divide__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),B2)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) ) ) ) ).

% divide_less_0_iff
tff(fact_2751_divide__less__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2752_divide__less__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2753_zero__less__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2754_zero__less__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),B2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2755_divide__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))) ) ) ).

% divide_strict_right_mono
tff(fact_2756_divide__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2))) ) ) ).

% divide_strict_right_mono
tff(fact_2757_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2758_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2759_right__inverse__eq,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) = one_one_complex )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2760_right__inverse__eq,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = one_one_real )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2761_right__inverse__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A2),B2) = one_one_rat )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2762_frac__eq__eq,axiom,
    ! [Y3: complex,Z: complex,X2: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z) )
        <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Z) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_2763_frac__eq__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_2764_frac__eq__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X2),Y3) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(X2),Z) = aa_rat_rat(times_times_rat(W),Y3) ) ) ) ) ).

% frac_eq_eq
tff(fact_2765_divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) = A2 )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_2766_divide__eq__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A2 )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_2767_divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C2) = A2 )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B2 = aa_rat_rat(times_times_rat(A2),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_2768_eq__divide__eq,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) = B2 ) )
        & ( ( C2 = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_2769_eq__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = B2 ) )
        & ( ( C2 = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_2770_eq__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A2),C2) = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_2771_divide__eq__imp,axiom,
    ! [C2: complex,B2: complex,A2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_2772_divide__eq__imp,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_2773_divide__eq__imp,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B2 = aa_rat_rat(times_times_rat(A2),C2) )
       => ( aa_rat_rat(divide_divide_rat(B2),C2) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_2774_eq__divide__imp,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) = B2 )
       => ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_2775_eq__divide__imp,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = B2 )
       => ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_2776_eq__divide__imp,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A2),C2) = B2 )
       => ( A2 = aa_rat_rat(divide_divide_rat(B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_2777_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B2: complex,A2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) = A2 )
      <=> ( B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2778_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A2 )
      <=> ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2779_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B2),C2) = A2 )
      <=> ( B2 = aa_rat_rat(times_times_rat(A2),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2780_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) )
      <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2781_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2782_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A2 = aa_rat_rat(divide_divide_rat(B2),C2) )
      <=> ( aa_rat_rat(times_times_rat(A2),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2783_field__le__epsilon,axiom,
    ! [X2: real,Y3: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y3),E2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% field_le_epsilon
tff(fact_2784_field__le__epsilon,axiom,
    ! [X2: rat,Y3: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_rat_rat(plus_plus_rat(Y3),E2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% field_le_epsilon
tff(fact_2785_frac__le,axiom,
    ! [Y3: real,X2: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y3),W))) ) ) ) ) ).

% frac_le
tff(fact_2786_frac__le,axiom,
    ! [Y3: rat,X2: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Z)),aa_rat_rat(divide_divide_rat(Y3),W))) ) ) ) ) ).

% frac_le
tff(fact_2787_frac__less,axiom,
    ! [X2: real,Y3: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y3),W))) ) ) ) ) ).

% frac_less
tff(fact_2788_frac__less,axiom,
    ! [X2: rat,Y3: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X2),Z)),aa_rat_rat(divide_divide_rat(Y3),W))) ) ) ) ) ).

% frac_less
tff(fact_2789_frac__less2,axiom,
    ! [X2: real,Y3: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y3),W))) ) ) ) ) ).

% frac_less2
tff(fact_2790_frac__less2,axiom,
    ! [X2: rat,Y3: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X2),Z)),aa_rat_rat(divide_divide_rat(Y3),W))) ) ) ) ) ).

% frac_less2
tff(fact_2791_divide__le__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ) ).

% divide_le_cancel
tff(fact_2792_divide__le__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ) ).

% divide_le_cancel
tff(fact_2793_divide__nonneg__neg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_2794_divide__nonneg__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_2795_divide__nonneg__pos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% divide_nonneg_pos
tff(fact_2796_divide__nonneg__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% divide_nonneg_pos
tff(fact_2797_divide__nonpos__neg,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% divide_nonpos_neg
tff(fact_2798_divide__nonpos__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% divide_nonpos_neg
tff(fact_2799_divide__nonpos__pos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_2800_divide__nonpos__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_2801_divide__less__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2802_divide__less__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2803_less__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ) ) ).

% less_divide_eq_1
tff(fact_2804_less__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ) ) ).

% less_divide_eq_1
tff(fact_2805_divide__less__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2806_divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2807_less__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2808_less__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2809_neg__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) ) ) ).

% neg_divide_less_eq
tff(fact_2810_neg__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) ) ) ).

% neg_divide_less_eq
tff(fact_2811_neg__less__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) ) ) ).

% neg_less_divide_eq
tff(fact_2812_neg__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) ) ) ).

% neg_less_divide_eq
tff(fact_2813_pos__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) ) ) ).

% pos_divide_less_eq
tff(fact_2814_pos__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) ) ) ).

% pos_divide_less_eq
tff(fact_2815_pos__less__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) ) ) ).

% pos_less_divide_eq
tff(fact_2816_pos__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) ) ) ).

% pos_less_divide_eq
tff(fact_2817_mult__imp__div__pos__less,axiom,
    ! [Y3: real,X2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2818_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(times_times_rat(Z),Y3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2819_mult__imp__less__div__pos,axiom,
    ! [Y3: real,Z: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3)),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2820_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y3)),X2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2821_divide__strict__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2822_divide__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2823_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2824_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2825_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),Z)),B2) = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),Z)),B2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2826_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B2) = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2827_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A2),Z)),B2) = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A2),Z)),B2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(times_times_rat(B2),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2828_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),Z)) = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),Z)),B2)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2829_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),Z)),B2)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2830_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(divide_divide_rat(B2),Z)) = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(divide_divide_rat(B2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),Z)),B2)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2831_add__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X2: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2832_add__frac__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2833_add__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),Y3)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2834_add__frac__num,axiom,
    ! [Y3: complex,X2: complex,Z: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3)),Z) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),Y3))),Y3) ) ) ).

% add_frac_num
tff(fact_2835_add__frac__num,axiom,
    ! [Y3: real,X2: real,Z: real] :
      ( ( Y3 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3))),Y3) ) ) ).

% add_frac_num
tff(fact_2836_add__frac__num,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),Y3)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(times_times_rat(Z),Y3))),Y3) ) ) ).

% add_frac_num
tff(fact_2837_add__num__frac,axiom,
    ! [Y3: complex,Z: complex,X2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),Y3))),Y3) ) ) ).

% add_num_frac
tff(fact_2838_add__num__frac,axiom,
    ! [Y3: real,Z: real,X2: real] :
      ( ( Y3 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3))),Y3) ) ) ).

% add_num_frac
tff(fact_2839_add__num__frac,axiom,
    ! [Y3: rat,Z: rat,X2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(X2),Y3)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(times_times_rat(Z),Y3))),Y3) ) ) ).

% add_num_frac
tff(fact_2840_add__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Y3),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Z)),Y3)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2841_add__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y3),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),Y3)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2842_add__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(divide_divide_rat(Y3),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),Z)),Y3)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2843_divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Z)),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2844_divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Z)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2845_divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),Z)),Y3) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(times_times_rat(Y3),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2846_gt__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B2)) ) ).

% gt_half_sum
tff(fact_2847_gt__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B2)) ) ).

% gt_half_sum
tff(fact_2848_less__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_2849_less__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)))) ) ).

% less_half_sum
tff(fact_2850_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),Z)) = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),Z)),B2)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2851_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),Z)),B2)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2852_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(divide_divide_rat(B2),Z)) = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(divide_divide_rat(B2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),Z)),B2)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2853_diff__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X2: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Y3)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2854_diff__frac__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2855_diff__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X2),Y3)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2856_diff__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Y3),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Z)),Y3)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2857_diff__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y3),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),Y3)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2858_diff__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(divide_divide_rat(Y3),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),Z)),Y3)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2859_divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,X2),Z)),Y3) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y3),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2860_divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Z)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2861_divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X2),Z)),Y3) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(Y3),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2862_field__le__mult__one__interval,axiom,
    ! [X2: real,Y3: real] :
      ( ! [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),X2)),Y3)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% field_le_mult_one_interval
tff(fact_2863_field__le__mult__one__interval,axiom,
    ! [X2: rat,Y3: rat] :
      ( ! [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X2)),Y3)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% field_le_mult_one_interval
tff(fact_2864_divide__le__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2865_divide__le__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2866_le__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ) ).

% le_divide_eq_1
tff(fact_2867_le__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ) ).

% le_divide_eq_1
tff(fact_2868_divide__le__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2869_divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2870_le__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2871_le__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2872_divide__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2))) ) ) ) ).

% divide_left_mono
tff(fact_2873_divide__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2))) ) ) ) ).

% divide_left_mono
tff(fact_2874_neg__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) ) ) ).

% neg_divide_le_eq
tff(fact_2875_neg__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) ) ) ).

% neg_divide_le_eq
tff(fact_2876_neg__le__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) ) ) ).

% neg_le_divide_eq
tff(fact_2877_neg__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) ) ) ).

% neg_le_divide_eq
tff(fact_2878_pos__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) ) ) ).

% pos_divide_le_eq
tff(fact_2879_pos__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2))) ) ) ).

% pos_divide_le_eq
tff(fact_2880_pos__le__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2)) ) ) ).

% pos_le_divide_eq
tff(fact_2881_pos__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2)) ) ) ).

% pos_le_divide_eq
tff(fact_2882_mult__imp__div__pos__le,axiom,
    ! [Y3: real,X2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2883_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_rat_rat(times_times_rat(Z),Y3)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2884_mult__imp__le__div__pos,axiom,
    ! [Y3: real,Z: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3)),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2885_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y3)),X2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X2),Y3))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2886_divide__left__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2887_divide__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2888_frac__le__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_2889_frac__le__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),aa_rat_rat(divide_divide_rat(W),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_2890_frac__less__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_2891_frac__less__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X2),Y3)),aa_rat_rat(divide_divide_rat(W),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(W),Y3))),aa_rat_rat(times_times_rat(Y3),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_2892_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),M))) ) ) ).

% less_log_of_power
tff(fact_2893_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_2894_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S2))),V)) ) ) ) ).

% scaling_mono
tff(fact_2895_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S2))),V)) ) ) ) ).

% scaling_mono
tff(fact_2896_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),M))) ) ) ).

% le_log_of_power
tff(fact_2897_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_2898_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_2899_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_2900_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_2901_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_2902_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,N)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% Tb_T_vebt_buildupi'
tff(fact_2903_vebt__buildupi__rule,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)),vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% vebt_buildupi_rule
tff(fact_2904_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_2905_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_2906_diff__gt__0__iff__gt,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_2907_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_2908_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_2909_diff__ge__0__iff__ge,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_2910_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_2911_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_2912_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,X2),Xa) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
             => ( ? [N3: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N3))
               => ( Y3 != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( Y3 = one_one_nat ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y3 = one_one_nat ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_2913_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2914_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2915_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2916_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2917_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2918_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2919_Leaf__0__not,axiom,
    ! [A2: bool,B2: bool] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_2920_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
    <=> ? [A5: bool,B5: bool] : T = aa_bool_VEBT_VEBT(vEBT_Leaf(A5),B5) ) ).

% deg1Leaf
tff(fact_2921_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) ) ).

% deg_1_Leaf
tff(fact_2922_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) ) ) ).

% deg_1_Leafy
tff(fact_2923_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N)) = aa_nat_int(vEBT_V9176841429113362141ildupi,N) ).

% Tbuildupi_buildupi'
tff(fact_2924_add__left__cancel,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2925_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2926_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2927_add__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2928_add__right__cancel,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2929_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = aa_rat_rat(plus_plus_rat(C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2930_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2931_add__right__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2932_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : ord_ma741700101516333627d_enat(Q3,zero_z5237406670263579293d_enat) = Q3 ).

% max_enat_simps(2)
tff(fact_2933_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_2934_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2935_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2936_add_Oright__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),zero_zero_complex) = A2 ).

% add.right_neutral
tff(fact_2937_add_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.right_neutral
tff(fact_2938_add_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),zero_zero_rat) = A2 ).

% add.right_neutral
tff(fact_2939_add_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.right_neutral
tff(fact_2940_add_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.right_neutral
tff(fact_2941_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_2942_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A2),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2943_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_2944_add__cancel__left__left,axiom,
    ! [B2: complex,A2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B2),A2) = A2 )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_2945_add__cancel__left__left,axiom,
    ! [B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = A2 )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_2946_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2947_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2948_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2949_add__cancel__left__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2) = A2 )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_2950_add__cancel__left__right,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = A2 )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_2951_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2952_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2953_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2954_add__cancel__right__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B2),A2) )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_2955_add__cancel__right__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_2956_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(B2),A2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2957_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2958_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2959_add__cancel__right__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2) )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_2960_add__cancel__right__right,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_2961_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(A2),B2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2962_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2963_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2964_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3) = zero_zero_nat )
    <=> ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2965_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3) )
    <=> ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2966_add__0,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A2) = A2 ).

% add_0
tff(fact_2967_add__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add_0
tff(fact_2968_add__0,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A2) = A2 ).

% add_0
tff(fact_2969_add__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% add_0
tff(fact_2970_add__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add_0
tff(fact_2971_add__le__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% add_le_cancel_left
tff(fact_2972_add__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% add_le_cancel_left
tff(fact_2973_add__le__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_cancel_left
tff(fact_2974_add__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_cancel_left
tff(fact_2975_add__le__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% add_le_cancel_right
tff(fact_2976_add__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% add_le_cancel_right
tff(fact_2977_add__le__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_cancel_right
tff(fact_2978_add__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_cancel_right
tff(fact_2979_add__less__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_2980_add__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_2981_add__less__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_2982_add__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_2983_add__less__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_2984_add__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_2985_add__less__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_2986_add__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_2987_diff__self,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),A2) = zero_zero_complex ).

% diff_self
tff(fact_2988_diff__self,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),A2) = zero_zero_real ).

% diff_self
tff(fact_2989_diff__self,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),A2) = zero_zero_rat ).

% diff_self
tff(fact_2990_diff__self,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% diff_self
tff(fact_2991_diff__0__right,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),zero_zero_complex) = A2 ).

% diff_0_right
tff(fact_2992_diff__0__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),zero_zero_real) = A2 ).

% diff_0_right
tff(fact_2993_diff__0__right,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),zero_zero_rat) = A2 ).

% diff_0_right
tff(fact_2994_diff__0__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_0_right
tff(fact_2995_zero__diff,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A2) = zero_zero_nat ).

% zero_diff
tff(fact_2996_diff__zero,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),zero_zero_complex) = A2 ).

% diff_zero
tff(fact_2997_diff__zero,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),zero_zero_real) = A2 ).

% diff_zero
tff(fact_2998_diff__zero,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),zero_zero_rat) = A2 ).

% diff_zero
tff(fact_2999_diff__zero,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),zero_zero_nat) = A2 ).

% diff_zero
tff(fact_3000_diff__zero,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_zero
tff(fact_3001_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),A2) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3002_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),A2) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3003_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),A2) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3004_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),A2) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3005_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3006_mult__1,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% mult_1
tff(fact_3007_mult__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A2) = A2 ).

% mult_1
tff(fact_3008_mult__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(one_one_rat),A2) = A2 ).

% mult_1
tff(fact_3009_mult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% mult_1
tff(fact_3010_mult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% mult_1
tff(fact_3011_mult_Oright__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% mult.right_neutral
tff(fact_3012_mult_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),one_one_real) = A2 ).

% mult.right_neutral
tff(fact_3013_mult_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),one_one_rat) = A2 ).

% mult.right_neutral
tff(fact_3014_mult_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.right_neutral
tff(fact_3015_mult_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.right_neutral
tff(fact_3016_add__diff__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_3017_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_3018_add__diff__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_3019_diff__add__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_3020_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_3021_diff__add__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_3022_add__diff__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) ).

% add_diff_cancel_left
tff(fact_3023_add__diff__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)) = aa_rat_rat(minus_minus_rat(A2),B2) ).

% add_diff_cancel_left
tff(fact_3024_add__diff__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ).

% add_diff_cancel_left
tff(fact_3025_add__diff__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) ).

% add_diff_cancel_left
tff(fact_3026_add__diff__cancel__left_H,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3027_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3028_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3029_add__diff__cancel__left_H,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3030_add__diff__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) ).

% add_diff_cancel_right
tff(fact_3031_add__diff__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(A2),B2) ).

% add_diff_cancel_right
tff(fact_3032_add__diff__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ).

% add_diff_cancel_right
tff(fact_3033_add__diff__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) ).

% add_diff_cancel_right
tff(fact_3034_add__diff__cancel__right_H,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3035_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3036_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3037_add__diff__cancel__right_H,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3038_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3039_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3040_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3041_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3042_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3043_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3044_le__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ).

% le_add_same_cancel2
tff(fact_3045_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(plus_plus_rat(B2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) ) ).

% le_add_same_cancel2
tff(fact_3046_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) ).

% le_add_same_cancel2
tff(fact_3047_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ).

% le_add_same_cancel2
tff(fact_3048_le__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2)) ) ).

% le_add_same_cancel1
tff(fact_3049_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(plus_plus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2)) ) ).

% le_add_same_cancel1
tff(fact_3050_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) ).

% le_add_same_cancel1
tff(fact_3051_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ).

% le_add_same_cancel1
tff(fact_3052_add__le__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_3053_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),B2)),B2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_3054_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_3055_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_3056_add__le__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_3057_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B2),A2)),B2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_3058_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_3059_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_3060_add__less__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_3061_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(B2),A2)),B2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_3062_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_3063_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_3064_add__less__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_3065_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),B2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_3066_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_3067_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_3068_less__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ).

% less_add_same_cancel1
tff(fact_3069_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) ) ).

% less_add_same_cancel1
tff(fact_3070_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ).

% less_add_same_cancel1
tff(fact_3071_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ).

% less_add_same_cancel1
tff(fact_3072_less__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2)) ) ).

% less_add_same_cancel2
tff(fact_3073_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(B2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2)) ) ).

% less_add_same_cancel2
tff(fact_3074_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ).

% less_add_same_cancel2
tff(fact_3075_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ).

% less_add_same_cancel2
tff(fact_3076_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3077_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_3078_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_3079_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,Ai: bool,Bi: bool] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),vEBT_Leafi(Ai,Bi)) = aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_bool_bool(fequal_bool(Ai),A2)),aa_bool_bool(fequal_bool(Bi),B2))) ).

% vebt_assn_raw.simps(1)
tff(fact_3080_VEBT_Oexhaust,axiom,
    ! [Y3: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142)
     => ~ ! [X212: bool,X223: bool] : Y3 != aa_bool_VEBT_VEBT(vEBT_Leaf(X212),X223) ) ).

% VEBT.exhaust
tff(fact_3081_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X222: bool] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != aa_bool_VEBT_VEBT(vEBT_Leaf(X21),X222) ).

% VEBT.distinct(1)
tff(fact_3082_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,D4: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_3083_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_3084_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_3085_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_3086_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,N: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2) ).

% vebt_delete.simps(3)
tff(fact_3087_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),zero_zero_nat) = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),B2) ).

% vebt_delete.simps(1)
tff(fact_3088_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_3089_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3090_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: bool,B2: bool] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_3091_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_3092_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: bool,B2: bool] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_VEBT_VEBT(vEBT_Leaf(A2),fFalse) ).

% vebt_delete.simps(2)
tff(fact_3093_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
     => ( ! [Uv2: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
       => ( ! [Uu2: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_3094_vebt__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2))
    <=> ( ( ( X2 = zero_zero_nat )
         => pp(A2) )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => pp(B2) )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_3095_vebt__buildup_Osimps_I2_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,zero_zero_nat)) = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ).

% vebt_buildup.simps(2)
tff(fact_3096_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
     => ( ! [Uv2: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
       => ( ! [Uu2: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3097_vebt__assn__raw_Ocases,axiom,
    ! [X2: produc3625547720036274456_VEBTi] :
      ( ! [A3: bool,B3: bool,Ai2: bool,Bi2: bool] : X2 != produc6084888613844515218_VEBTi(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),vEBT_Leafi(Ai2,Bi2))
     => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] : X2 != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))
       => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd: bool,Ve: bool] : X2 != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va3,Vb3),Vc3),vEBT_Leafi(Vd,Ve))
         => ~ ! [Vd: bool,Ve: bool,V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : X2 != produc6084888613844515218_VEBTi(aa_bool_VEBT_VEBT(vEBT_Leaf(Vd),Ve),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va3,Vb3),Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_3098_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A2: bool,B2: bool] :
      ( ( ( X2 = zero_zero_nat )
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),B2) ) )
      & ( ( X2 != zero_zero_nat )
       => ( ( ( X2 = one_one_nat )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = aa_bool_VEBT_VEBT(vEBT_Leaf(A2),fTrue) ) )
          & ( ( X2 != one_one_nat )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_3099_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3100_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3101_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3102_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3103_group__cancel_Oadd1,axiom,
    ! [A: real,K: real,A2: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3104_group__cancel_Oadd1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),A2) )
     => ( aa_rat_rat(plus_plus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3105_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,A2: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3106_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3107_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,B2: real,A2: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_3108_group__cancel_Oadd2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(K),B2) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_3109_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A2: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_3110_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_3111_add_Oassoc,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% add.assoc
tff(fact_3112_add_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% add.assoc
tff(fact_3113_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% add.assoc
tff(fact_3114_add_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% add.assoc
tff(fact_3115_add_Oleft__cancel,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_3116_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_3117_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_3118_add_Oright__cancel,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_3119_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = aa_rat_rat(plus_plus_rat(C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_3120_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_3121_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3122_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3123_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3124_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3125_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: real,A2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3126_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(plus_plus_rat(A2),C2)) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3127_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3128_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3129_add__left__imp__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_3130_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_3131_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_3132_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_3133_add__right__imp__eq,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_3134_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = aa_rat_rat(plus_plus_rat(C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_3135_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_3136_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_3137_one__reorient,axiom,
    ! [X2: assn] :
      ( ( one_one_assn = X2 )
    <=> ( X2 = one_one_assn ) ) ).

% one_reorient
tff(fact_3138_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
    <=> ( X2 = one_one_real ) ) ).

% one_reorient
tff(fact_3139_one__reorient,axiom,
    ! [X2: rat] :
      ( ( one_one_rat = X2 )
    <=> ( X2 = one_one_rat ) ) ).

% one_reorient
tff(fact_3140_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
    <=> ( X2 = one_one_nat ) ) ).

% one_reorient
tff(fact_3141_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
    <=> ( X2 = one_one_int ) ) ).

% one_reorient
tff(fact_3142_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: real,A2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3143_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] : aa_rat_rat(times_times_rat(B2),aa_rat_rat(times_times_rat(A2),C2)) = aa_rat_rat(times_times_rat(A2),aa_rat_rat(times_times_rat(B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3144_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3145_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3146_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3147_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(A2),B2) = aa_rat_rat(times_times_rat(B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3148_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3149_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3150_mult_Oassoc,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% mult.assoc
tff(fact_3151_mult_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A2),B2)),C2) = aa_rat_rat(times_times_rat(A2),aa_rat_rat(times_times_rat(B2),C2)) ).

% mult.assoc
tff(fact_3152_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% mult.assoc
tff(fact_3153_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) ).

% mult.assoc
tff(fact_3154_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
     => ( ( X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3155_diff__eq__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2) )
     => ( ( A2 = B2 )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_3156_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(minus_minus_rat(C2),D2) )
     => ( ( A2 = B2 )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_3157_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( ( A2 = B2 )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_3158_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: real,C2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3159_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),C2)),B2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3160_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3161_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C2)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3162_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_3163_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_3164_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_3165_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),zero_zero_nat) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_3166_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A3: bool,B3: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_3167_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A2: bool,B2: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_3168_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A2: bool,B2: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_3169_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y3: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
      <=> pp(Y3) )
     => ( ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
         => ~ pp(Y3) )
       => ( ( ? [Uv2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
           => pp(Y3) )
         => ( ( ? [Uu2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
             => pp(Y3) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ~ pp(Y3) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2)
                 => pp(Y3) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3170_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_3171_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X2) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_3172_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),zero_zero_nat)
     => ( ! [A3: bool,B3: bool] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B3: bool,N3: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,N3)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),X4)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),X4)
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X4) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_3173_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),X4)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X4) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_3174_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3),X4)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X4) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_3175_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B3: bool] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3),zero_zero_nat)
     => ( ! [Uv2: bool,Uw2: bool,N3: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2),aa_nat_nat(suc,N3))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Va2)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Vi)
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X4) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_3176_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),zero_zero_nat)
     => ( ! [A3: bool,Uw2: bool] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B3: bool,Va3: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Vf)
             => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Vj)
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),X4) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_3177_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X2 != produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),X4)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),X4)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3178_zero__le,axiom,
    ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2)) ).

% zero_le
tff(fact_3179_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_3180_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_3181_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_3182_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_3183_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3184_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3185_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3186_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3187_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3188_add_Ocomm__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),zero_zero_complex) = A2 ).

% add.comm_neutral
tff(fact_3189_add_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.comm_neutral
tff(fact_3190_add_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),zero_zero_rat) = A2 ).

% add.comm_neutral
tff(fact_3191_add_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.comm_neutral
tff(fact_3192_add_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.comm_neutral
tff(fact_3193_add_Ogroup__left__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A2) = A2 ).

% add.group_left_neutral
tff(fact_3194_add_Ogroup__left__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add.group_left_neutral
tff(fact_3195_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A2) = A2 ).

% add.group_left_neutral
tff(fact_3196_add_Ogroup__left__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add.group_left_neutral
tff(fact_3197_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3198_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3199_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3200_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3201_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3202_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3203_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3204_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3205_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3206_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3207_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3208_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3209_add__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2))) ) ) ).

% add_mono
tff(fact_3210_add__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2))) ) ) ).

% add_mono
tff(fact_3211_add__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_mono
tff(fact_3212_add__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_mono
tff(fact_3213_add__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))) ) ).

% add_left_mono
tff(fact_3214_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2))) ) ).

% add_left_mono
tff(fact_3215_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))) ) ).

% add_left_mono
tff(fact_3216_add__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))) ) ).

% add_left_mono
tff(fact_3217_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ~ ! [C3: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C3) ) ).

% less_eqE
tff(fact_3218_add__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))) ) ).

% add_right_mono
tff(fact_3219_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2))) ) ).

% add_right_mono
tff(fact_3220_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))) ) ).

% add_right_mono
tff(fact_3221_add__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))) ) ).

% add_right_mono
tff(fact_3222_le__iff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ? [C4: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C4) ) ).

% le_iff_add
tff(fact_3223_add__le__imp__le__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3224_add__le__imp__le__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3225_add__le__imp__le__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3226_add__le__imp__le__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3227_add__le__imp__le__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3228_add__le__imp__le__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3229_add__le__imp__le__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3230_add__le__imp__le__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3231_add__less__imp__less__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3232_add__less__imp__less__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3233_add__less__imp__less__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3234_add__less__imp__less__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3235_add__less__imp__less__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3236_add__less__imp__less__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3237_add__less__imp__less__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3238_add__less__imp__less__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3239_add__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))) ) ).

% add_strict_right_mono
tff(fact_3240_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2))) ) ).

% add_strict_right_mono
tff(fact_3241_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))) ) ).

% add_strict_right_mono
tff(fact_3242_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))) ) ).

% add_strict_right_mono
tff(fact_3243_add__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))) ) ).

% add_strict_left_mono
tff(fact_3244_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2))) ) ).

% add_strict_left_mono
tff(fact_3245_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))) ) ).

% add_strict_left_mono
tff(fact_3246_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))) ) ).

% add_strict_left_mono
tff(fact_3247_add__strict__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3248_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3249_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3250_add__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3251_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3252_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3253_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3254_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3255_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3256_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3257_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3258_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3259_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3260_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3261_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3262_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3263_eq__iff__diff__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = B2 )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_3264_eq__iff__diff__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3265_eq__iff__diff__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( aa_rat_rat(minus_minus_rat(A2),B2) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3266_eq__iff__diff__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3267_diff__mono,axiom,
    ! [A2: real,B2: real,D2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2))) ) ) ).

% diff_mono
tff(fact_3268_diff__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),D2))) ) ) ).

% diff_mono
tff(fact_3269_diff__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D2))) ) ) ).

% diff_mono
tff(fact_3270_diff__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2))) ) ).

% diff_left_mono
tff(fact_3271_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C2),A2)),aa_rat_rat(minus_minus_rat(C2),B2))) ) ).

% diff_left_mono
tff(fact_3272_diff__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2))) ) ).

% diff_left_mono
tff(fact_3273_diff__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2))) ) ).

% diff_right_mono
tff(fact_3274_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),C2))) ) ).

% diff_right_mono
tff(fact_3275_diff__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2))) ) ).

% diff_right_mono
tff(fact_3276_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3277_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(minus_minus_rat(C2),D2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3278_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3279_diff__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2))) ) ).

% diff_strict_right_mono
tff(fact_3280_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),C2))) ) ).

% diff_strict_right_mono
tff(fact_3281_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2))) ) ).

% diff_strict_right_mono
tff(fact_3282_diff__strict__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2))) ) ).

% diff_strict_left_mono
tff(fact_3283_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(C2),A2)),aa_rat_rat(minus_minus_rat(C2),B2))) ) ).

% diff_strict_left_mono
tff(fact_3284_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2))) ) ).

% diff_strict_left_mono
tff(fact_3285_diff__eq__diff__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2)) ) ) ).

% diff_eq_diff_less
tff(fact_3286_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(minus_minus_rat(C2),D2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2)) ) ) ).

% diff_eq_diff_less
tff(fact_3287_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2)) ) ) ).

% diff_eq_diff_less
tff(fact_3288_diff__strict__mono,axiom,
    ! [A2: real,B2: real,D2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2))) ) ) ).

% diff_strict_mono
tff(fact_3289_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),D2))) ) ) ).

% diff_strict_mono
tff(fact_3290_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D2))) ) ) ).

% diff_strict_mono
tff(fact_3291_mult_Ocomm__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% mult.comm_neutral
tff(fact_3292_mult_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),one_one_real) = A2 ).

% mult.comm_neutral
tff(fact_3293_mult_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),one_one_rat) = A2 ).

% mult.comm_neutral
tff(fact_3294_mult_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.comm_neutral
tff(fact_3295_mult_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.comm_neutral
tff(fact_3296_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_3297_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_3298_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(one_one_rat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_3299_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_3300_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_3301_group__cancel_Osub1,axiom,
    ! [A: real,K: real,A2: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_3302_group__cancel_Osub1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),A2) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_3303_group__cancel_Osub1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_3304_diff__eq__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = C2 )
    <=> ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2) ) ) ).

% diff_eq_eq
tff(fact_3305_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = C2 )
    <=> ( A2 = aa_rat_rat(plus_plus_rat(C2),B2) ) ) ).

% diff_eq_eq
tff(fact_3306_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = C2 )
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2) ) ) ).

% diff_eq_eq
tff(fact_3307_eq__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_3308_eq__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(minus_minus_rat(C2),B2) )
    <=> ( aa_rat_rat(plus_plus_rat(A2),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_3309_eq__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_3310_add__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) ).

% add_diff_eq
tff(fact_3311_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) ).

% add_diff_eq
tff(fact_3312_add__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ).

% add_diff_eq
tff(fact_3313_diff__diff__eq2,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2) ).

% diff_diff_eq2
tff(fact_3314_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),B2) ).

% diff_diff_eq2
tff(fact_3315_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),B2) ).

% diff_diff_eq2
tff(fact_3316_diff__add__eq,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2) ).

% diff_add_eq
tff(fact_3317_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),B2) ).

% diff_add_eq
tff(fact_3318_diff__add__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),B2) ).

% diff_add_eq
tff(fact_3319_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3320_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3321_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3322_add__implies__diff,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2) = A2 )
     => ( C2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) ) ) ).

% add_implies_diff
tff(fact_3323_add__implies__diff,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C2),B2) = A2 )
     => ( C2 = aa_rat_rat(minus_minus_rat(A2),B2) ) ) ).

% add_implies_diff
tff(fact_3324_add__implies__diff,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2) = A2 )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ) ) ).

% add_implies_diff
tff(fact_3325_add__implies__diff,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2) = A2 )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) ) ) ).

% add_implies_diff
tff(fact_3326_diff__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% diff_diff_eq
tff(fact_3327_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% diff_diff_eq
tff(fact_3328_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% diff_diff_eq
tff(fact_3329_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)) ).

% diff_diff_eq
tff(fact_3330_max__add__distrib__left,axiom,
    ! [X2: real,Y3: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y3)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y3),Z)) ).

% max_add_distrib_left
tff(fact_3331_max__add__distrib__left,axiom,
    ! [X2: rat,Y3: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(X2),Y3)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X2),Z)),aa_rat_rat(plus_plus_rat(Y3),Z)) ).

% max_add_distrib_left
tff(fact_3332_max__add__distrib__left,axiom,
    ! [X2: nat,Y3: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),Z)) ).

% max_add_distrib_left
tff(fact_3333_max__add__distrib__left,axiom,
    ! [X2: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),Z)) ).

% max_add_distrib_left
tff(fact_3334_max__add__distrib__right,axiom,
    ! [X2: real,Y3: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y3),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Z)) ).

% max_add_distrib_right
tff(fact_3335_max__add__distrib__right,axiom,
    ! [X2: rat,Y3: rat,Z: rat] : aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(ord_max_rat(Y3),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X2),Y3)),aa_rat_rat(plus_plus_rat(X2),Z)) ).

% max_add_distrib_right
tff(fact_3336_max__add__distrib__right,axiom,
    ! [X2: nat,Y3: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y3),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Z)) ).

% max_add_distrib_right
tff(fact_3337_max__add__distrib__right,axiom,
    ! [X2: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(ord_max_int,Y3),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Z)) ).

% max_add_distrib_right
tff(fact_3338_max__diff__distrib__left,axiom,
    ! [X2: real,Y3: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y3)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),Z)) ).

% max_diff_distrib_left
tff(fact_3339_max__diff__distrib__left,axiom,
    ! [X2: rat,Y3: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X2),Y3)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X2),Z)),aa_rat_rat(minus_minus_rat(Y3),Z)) ).

% max_diff_distrib_left
tff(fact_3340_max__diff__distrib__left,axiom,
    ! [X2: int,Y3: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y3),Z)) ).

% max_diff_distrib_left
tff(fact_3341_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_3342_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),X2) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_3343_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X2) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_3344_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% member_bound_height'
tff(fact_3345_add__nonpos__eq__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3) = zero_zero_real )
        <=> ( ( X2 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3346_add__nonpos__eq__0__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),zero_zero_rat))
       => ( ( aa_rat_rat(plus_plus_rat(X2),Y3) = zero_zero_rat )
        <=> ( ( X2 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3347_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3) = zero_zero_nat )
        <=> ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3348_add__nonpos__eq__0__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3) = zero_zero_int )
        <=> ( ( X2 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3349_add__nonneg__eq__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3) = zero_zero_real )
        <=> ( ( X2 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3350_add__nonneg__eq__0__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y3))
       => ( ( aa_rat_rat(plus_plus_rat(X2),Y3) = zero_zero_rat )
        <=> ( ( X2 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3351_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y3))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3) = zero_zero_nat )
        <=> ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3352_add__nonneg__eq__0__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3) = zero_zero_int )
        <=> ( ( X2 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3353_add__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_3354_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_3355_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_3356_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_3357_add__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_3358_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_3359_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_3360_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_3361_add__increasing2,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2))) ) ) ).

% add_increasing2
tff(fact_3362_add__increasing2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2))) ) ) ).

% add_increasing2
tff(fact_3363_add__increasing2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2))) ) ) ).

% add_increasing2
tff(fact_3364_add__increasing2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2))) ) ) ).

% add_increasing2
tff(fact_3365_add__decreasing2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2)) ) ) ).

% add_decreasing2
tff(fact_3366_add__decreasing2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),B2)) ) ) ).

% add_decreasing2
tff(fact_3367_add__decreasing2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),B2)) ) ) ).

% add_decreasing2
tff(fact_3368_add__decreasing2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),B2)) ) ) ).

% add_decreasing2
tff(fact_3369_add__increasing,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2))) ) ) ).

% add_increasing
tff(fact_3370_add__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2))) ) ) ).

% add_increasing
tff(fact_3371_add__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2))) ) ) ).

% add_increasing
tff(fact_3372_add__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2))) ) ) ).

% add_increasing
tff(fact_3373_add__decreasing,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2)) ) ) ).

% add_decreasing
tff(fact_3374_add__decreasing,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),B2)) ) ) ).

% add_decreasing
tff(fact_3375_add__decreasing,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),B2)) ) ) ).

% add_decreasing
tff(fact_3376_add__decreasing,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),B2)) ) ) ).

% add_decreasing
tff(fact_3377_add__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_3378_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_3379_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_3380_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_3381_add__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))) ) ) ).

% add_pos_pos
tff(fact_3382_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2))) ) ) ).

% add_pos_pos
tff(fact_3383_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_pos_pos
tff(fact_3384_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_pos_pos
tff(fact_3385_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ~ ! [C3: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3386_pos__add__strict,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2))) ) ) ).

% pos_add_strict
tff(fact_3387_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2))) ) ) ).

% pos_add_strict
tff(fact_3388_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2))) ) ) ).

% pos_add_strict
tff(fact_3389_pos__add__strict,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2))) ) ) ).

% pos_add_strict
tff(fact_3390_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3391_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3392_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3393_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3394_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3395_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3396_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3397_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3398_add__le__less__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3399_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3400_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3401_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3402_add__less__le__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3403_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3404_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3405_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3406_le__iff__diff__le__0,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_3407_le__iff__diff__le__0,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),B2)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_3408_le__iff__diff__le__0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_3409_less__iff__diff__less__0,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_3410_less__iff__diff__less__0,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),B2)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_3411_less__iff__diff__less__0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_3412_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2) = C2 )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3413_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3414_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3415_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3416_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3417_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3418_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3419_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),B2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3420_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)),A2))) ) ).

% le_add_diff
tff(fact_3421_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),A2) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_3422_le__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2)) ) ).

% le_diff_eq
tff(fact_3423_le__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(minus_minus_rat(C2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),B2)),C2)) ) ).

% le_diff_eq
tff(fact_3424_le__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2)) ) ).

% le_diff_eq
tff(fact_3425_diff__le__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))) ) ).

% diff_le_eq
tff(fact_3426_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),B2)),C2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(plus_plus_rat(C2),B2))) ) ).

% diff_le_eq
tff(fact_3427_diff__le__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))) ) ).

% diff_le_eq
tff(fact_3428_diff__less__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))) ) ).

% diff_less_eq
tff(fact_3429_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),B2)),C2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(C2),B2))) ) ).

% diff_less_eq
tff(fact_3430_diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B2))) ) ).

% diff_less_eq
tff(fact_3431_less__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2)) ) ).

% less_diff_eq
tff(fact_3432_less__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(minus_minus_rat(C2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),C2)) ) ).

% less_diff_eq
tff(fact_3433_less__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2)) ) ).

% less_diff_eq
tff(fact_3434_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,X2),Xa) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => ( Y3 != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
                   => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_3435_add__neg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_3436_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_3437_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_3438_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_3439_add__nonneg__pos,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_3440_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_3441_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_3442_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_3443_add__nonpos__neg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_3444_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_3445_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_3446_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_3447_add__pos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_3448_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_3449_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_3450_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_3451_add__strict__increasing,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2))) ) ) ).

% add_strict_increasing
tff(fact_3452_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2))) ) ) ).

% add_strict_increasing
tff(fact_3453_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2))) ) ) ).

% add_strict_increasing
tff(fact_3454_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2))) ) ) ).

% add_strict_increasing
tff(fact_3455_add__strict__increasing2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2))) ) ) ).

% add_strict_increasing2
tff(fact_3456_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2))) ) ) ).

% add_strict_increasing2
tff(fact_3457_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2))) ) ) ).

% add_strict_increasing2
tff(fact_3458_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2))) ) ) ).

% add_strict_increasing2
tff(fact_3459_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => pp(B3) )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3460_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa))
      <=> pp(Y3) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ( pp(Y3)
            <=> ~ ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => pp(Y3) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => pp(Y3) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => pp(Y3) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
                   => ( pp(Y3)
                    <=> ~ ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3461_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ( ( ( Xa = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => pp(B3) )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3462_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,X2),Xa) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)
                   => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_3463_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,X2),Xa) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)
               => ( Y3 != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                   => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_3464_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
    <=> ( ( ? [A5: bool,B5: bool] : A1 = aa_bool_VEBT_VEBT(vEBT_Leaf(A5),B5)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N7))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),N7) )
            & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_1))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N7)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N7)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),aa_nat_nat(suc,N7)) )
            & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_1))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N7))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),N7) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))
               => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I4)),X_1))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N7) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N7))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N7) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N7))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N7)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N7)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),aa_nat_nat(suc,N7)) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N7))))
               => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I4)),X_1))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N7))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N7) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N7))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N7) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N7))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3465_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
     => ( ( ? [A3: bool,B3: bool] : A1 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X6: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X6),N3)) )
                 => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M5))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5) )
                     => ( ( M5 = N3 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M5) )
                         => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                           => ~ ! [X6: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X6),X_13)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X6),N3)) )
                   => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M5))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5) )
                       => ( ( M5 = aa_nat_nat(suc,N3) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M5) )
                           => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                             => ~ ! [X6: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X6),X_13)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X6: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X6),N3)) )
                     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M5))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5) )
                         => ( ( M5 = N3 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M5) )
                             => ( ! [I5: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I5)),X_1))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I5)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X6: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X6),X_13)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I5: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I5 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3))) )
                                                & ! [X6: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X6),N3) = I5 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X6),N3))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X6))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X6: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X6),N3)) )
                       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M5))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5) )
                           => ( ( M5 = aa_nat_nat(suc,N3) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M5) )
                               => ( ! [I5: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                     => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I5)),X_1))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I5)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X6: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X6),X_13)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I5: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I5 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3))) )
                                                  & ! [X6: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X6),N3) = I5 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X6),N3))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X6))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3466_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(X2),Xa) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),B3) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fTrue) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
             => ( Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
               => ( Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                 => ( Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                   => ( Y3 != aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3467_vebt__delete_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(X2),Xa) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ( ( Xa = zero_zero_nat )
             => ( Y3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),B3) ) ) )
       => ( ! [A3: bool] :
              ( ? [B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
             => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y3 != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fFalse) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
               => ( ? [N3: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N3))
                 => ( Y3 != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                 => ( Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                     => ( Y3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y3 = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),a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um_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_3468_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_VEBT_minNulli(Ti),aa_VEB1972364096475303967l_assn(aTP_Lamm_bl(T),Ti)),one_one_nat)) ).

% minNrulli_ruleT
tff(fact_3469_vebt__maxt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_maxt(X2) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ~ ( ( pp(B3)
                 => ( Y3 = some_nat(one_one_nat) ) )
                & ( ~ pp(B3)
                 => ( ( pp(A3)
                     => ( Y3 = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y3 != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3470_vebt__mint_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_mint(X2) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ~ ( ( pp(A3)
                 => ( Y3 = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A3)
                 => ( ( pp(B3)
                     => ( Y3 = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y3 != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3471_norm__pre__pure__iff__sng,axiom,
    ! [B2: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_bool_assn(pure_assn,B2),F,Q)
    <=> ( pp(B2)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3472_norm__pre__pure__iff__sng,axiom,
    ! [B2: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(aa_bool_assn(pure_assn,B2),F,Q)
    <=> ( pp(B2)
       => hoare_2774773592644289642e_bool(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3473_norm__pre__pure__iff__sng,axiom,
    ! [B2: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_bool_assn(pure_assn,B2),F,Q)
    <=> ( pp(B2)
       => hoare_7629718768684598413on_nat(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3474_norm__pre__pure__iff__sng,axiom,
    ! [B2: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_bool_assn(pure_assn,B2),F,Q)
    <=> ( pp(B2)
       => hoare_3067605981109127869le_nat(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3475_ent__pure__pre__iff__sng,axiom,
    ! [B2: bool,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_bool_assn(pure_assn,B2)),Q))
    <=> ( pp(B2)
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,one_one_assn),Q)) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_3476_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q)
    <=> ( pp(B2)
       => hoare_1429296392585015714_VEBTi(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3477_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q)
    <=> ( pp(B2)
       => hoare_2774773592644289642e_bool(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3478_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q)
    <=> ( pp(B2)
       => hoare_7629718768684598413on_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3479_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q)
    <=> ( pp(B2)
       => hoare_3067605981109127869le_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3480_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: bool,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2))),Q))
    <=> ( pp(B2)
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ).

% ent_pure_pre_iff
tff(fact_3481_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: bool,A2: bool,Va: nat] :
      ( ( pp(B2)
       => ( vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(one_one_nat) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_3482_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_2774773592644289642e_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_VEBT_minNulli(Ti),aa_VEB1972364096475303967l_assn(aTP_Lamm_bl(T),Ti)) ).

% minNulli_rule
tff(fact_3483_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : hoare_2774773592644289642e_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_V854960066525838166emberi(T,Ti,X2),aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_bm(T),Ti),X2)) ).

% vebt_memberi'_rf_abstr
tff(fact_3484_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_VEBT_vebt_predi(T,Ti,X2),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_bn(T),Ti),X2)) ) ).

% vebt_pred'_rf_abstr
tff(fact_3485_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_VEBT_vebt_succi(T,Ti,X2),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_bo(T),Ti),X2)) ) ).

% vebt_succi'_rf_abstr
tff(fact_3486_merge__pure__star,axiom,
    ! [A2: bool,B2: bool] : aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,A2)),aa_bool_assn(pure_assn,B2)) = aa_bool_assn(pure_assn,aa_bool_bool(fconj(A2),B2)) ).

% merge_pure_star
tff(fact_3487_pure__true,axiom,
    aa_bool_assn(pure_assn,fTrue) = one_one_assn ).

% pure_true
tff(fact_3488_pure__assn__eq__emp__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_assn(pure_assn,P) = one_one_assn )
    <=> pp(P) ) ).

% pure_assn_eq_emp_iff
tff(fact_3489_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : pp(aa_nat_bool(time_htt_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_memberi(Ti,X2),aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_bm(T),Ti),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% htt_vebt_memberi
tff(fact_3490_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(times_times_assn(P),Q) = aa_assn_assn(times_times_assn(Q),P) ).

% assn_times_comm
tff(fact_3491_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),R) = aa_assn_assn(times_times_assn(P),aa_assn_assn(times_times_assn(Q),R)) ).

% assn_times_assoc
tff(fact_3492_ent__iffI,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),B))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),A))
       => ( A = B ) ) ) ).

% ent_iffI
tff(fact_3493_ent__refl,axiom,
    ! [P: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P)) ).

% ent_refl
tff(fact_3494_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Q),R))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R)) ) ) ).

% ent_trans
tff(fact_3495_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X2 != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X2 != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] : X2 != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3496_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X2 != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : X2 != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat] : X2 != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3497_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc533602882674032673at_nat] :
      ( ! [Uu2: fun_Pr248683899324676269t_bool,Uv2: option4927543243414619207at_nat] : X2 != produc5961576194287291153at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr248683899324676269t_bool,V2: product_prod_nat_nat] : X2 != produc5961576194287291153at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr248683899324676269t_bool,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : X2 != produc5961576194287291153at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X4),some_P7363390416028606310at_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3498_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc3700678068921409119on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_bool,Uv2: option_nat] : X2 != produc2004789812759622991on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_bool,V2: nat] : X2 != produc2004789812759622991on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_bool,X4: nat,Y4: nat] : X2 != produc2004789812759622991on_nat(F2,produc5098337634421038937on_nat(some_nat(X4),some_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3499_ent__star__mono,axiom,
    ! [P: assn,P3: assn,Q: assn,Q2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Q),Q2))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(P),Q)),aa_assn_assn(times_times_assn(P3),Q2))) ) ) ).

% ent_star_mono
tff(fact_3500_cons__rule,axiom,
    ! [P: assn,P3: assn,Q: fun_VEBT_VEBTi_assn,Q2: fun_VEBT_VEBTi_assn,C2: heap_T8145700208782473153_VEBTi] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( ! [X4: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q,X4)),aa_VEBT_VEBTi_assn(Q2,X4)))
       => ( hoare_1429296392585015714_VEBTi(P3,C2,Q)
         => hoare_1429296392585015714_VEBTi(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3501_cons__rule,axiom,
    ! [P: assn,P3: assn,Q: fun_bool_assn,Q2: fun_bool_assn,C2: heap_Time_Heap_bool] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( ! [X4: bool] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_bool_assn(Q,X4)),aa_bool_assn(Q2,X4)))
       => ( hoare_2774773592644289642e_bool(P3,C2,Q)
         => hoare_2774773592644289642e_bool(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3502_cons__rule,axiom,
    ! [P: assn,P3: assn,Q: fun_option_nat_assn,Q2: fun_option_nat_assn,C2: heap_T2636463487746394924on_nat] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( ! [X4: option_nat] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_option_nat_assn(Q,X4)),aa_option_nat_assn(Q2,X4)))
       => ( hoare_7629718768684598413on_nat(P3,C2,Q)
         => hoare_7629718768684598413on_nat(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3503_cons__rule,axiom,
    ! [P: assn,P3: assn,Q: fun_nat_assn,Q2: fun_nat_assn,C2: heap_Time_Heap_nat] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( ! [X4: nat] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_nat_assn(Q,X4)),aa_nat_assn(Q2,X4)))
       => ( hoare_3067605981109127869le_nat(P3,C2,Q)
         => hoare_3067605981109127869le_nat(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3504_cons__post__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Q2: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => ( ! [X4: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q,X4)),aa_VEBT_VEBTi_assn(Q2,X4)))
       => hoare_1429296392585015714_VEBTi(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3505_cons__post__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn,Q2: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(P,C2,Q)
     => ( ! [X4: bool] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_bool_assn(Q,X4)),aa_bool_assn(Q2,X4)))
       => hoare_2774773592644289642e_bool(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3506_cons__post__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Q2: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => ( ! [X4: option_nat] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_option_nat_assn(Q,X4)),aa_option_nat_assn(Q2,X4)))
       => hoare_7629718768684598413on_nat(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3507_cons__post__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,Q2: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => ( ! [X4: nat] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_nat_assn(Q,X4)),aa_nat_assn(Q2,X4)))
       => hoare_3067605981109127869le_nat(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3508_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(one_one_assn),P) = P ).

% assn_one_left
tff(fact_3509_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B2)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_3510_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: nat,B2: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A2)),some_nat(B2)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_3511_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3512_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_3513_frame__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),R),C2,aa_ass4389913640333755968i_assn(aTP_Lamm_bp(Q),R)) ) ).

% frame_rule
tff(fact_3514_frame__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn,R: assn] :
      ( hoare_2774773592644289642e_bool(P,C2,Q)
     => hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(P),R),C2,aa_ass2376637134808068204l_assn(aTP_Lamm_bq(Q),R)) ) ).

% frame_rule
tff(fact_3515_frame__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),R),C2,aa_ass4112278883933122603t_assn(aTP_Lamm_br(Q),R)) ) ).

% frame_rule
tff(fact_3516_frame__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),R),C2,aa_assn_fun_nat_assn(aTP_Lamm_bs(Q),R)) ) ).

% frame_rule
tff(fact_3517_cons__pre__rule,axiom,
    ! [P: assn,P3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( hoare_1429296392585015714_VEBTi(P3,C2,Q)
       => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_3518_cons__pre__rule,axiom,
    ! [P: assn,P3: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( hoare_2774773592644289642e_bool(P3,C2,Q)
       => hoare_2774773592644289642e_bool(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_3519_cons__pre__rule,axiom,
    ! [P: assn,P3: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( hoare_7629718768684598413on_nat(P3,C2,Q)
       => hoare_7629718768684598413on_nat(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_3520_cons__pre__rule,axiom,
    ! [P: assn,P3: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
     => ( hoare_3067605981109127869le_nat(P3,C2,Q)
       => hoare_3067605981109127869le_nat(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_3521_norm__pre__pure__rule1,axiom,
    ! [B2: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( pp(B2)
       => hoare_1429296392585015714_VEBTi(P,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3522_norm__pre__pure__rule1,axiom,
    ! [B2: bool,P: assn,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( ( pp(B2)
       => hoare_2774773592644289642e_bool(P,F,Q) )
     => hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3523_norm__pre__pure__rule1,axiom,
    ! [B2: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( pp(B2)
       => hoare_7629718768684598413on_nat(P,F,Q) )
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3524_norm__pre__pure__rule1,axiom,
    ! [B2: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( pp(B2)
       => hoare_3067605981109127869le_nat(P,F,Q) )
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3525_norm__pre__pure__rule2,axiom,
    ! [B2: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( pp(B2)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_bool_assn(pure_assn,B2),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3526_norm__pre__pure__rule2,axiom,
    ! [B2: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( ( pp(B2)
       => hoare_2774773592644289642e_bool(one_one_assn,F,Q) )
     => hoare_2774773592644289642e_bool(aa_bool_assn(pure_assn,B2),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3527_norm__pre__pure__rule2,axiom,
    ! [B2: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( pp(B2)
       => hoare_7629718768684598413on_nat(one_one_assn,F,Q) )
     => hoare_7629718768684598413on_nat(aa_bool_assn(pure_assn,B2),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3528_norm__pre__pure__rule2,axiom,
    ! [B2: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( pp(B2)
       => hoare_3067605981109127869le_nat(one_one_assn,F,Q) )
     => hoare_3067605981109127869le_nat(aa_bool_assn(pure_assn,B2),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3529_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3530_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_3531_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: fun_Pr2477239776880934807at_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X2,Xa,Xb) = Y3 )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y3 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xa = some_P7363390416028606310at_nat(V2)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y3 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa = some_P7363390416028606310at_nat(A3) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb = some_P7363390416028606310at_nat(B3) )
                   => ( Y3 != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X2,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3532_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: fun_nat_fun_nat_nat,Xa: option_nat,Xb: option_nat,Y3: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X2),Xa),Xb) = Y3 )
     => ( ( ( Xa = none_nat )
         => ( Y3 != none_nat ) )
       => ( ( ? [V2: nat] : Xa = some_nat(V2)
           => ( ( Xb = none_nat )
             => ( Y3 != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa = some_nat(A3) )
               => ! [B3: nat] :
                    ( ( Xb = some_nat(B3) )
                   => ( Y3 != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X2,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3533_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X2),Y3))
    <=> ( pp(aa_set_nat_bool(member_nat(Y3),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_3534_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_3535_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_3536_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_3537_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_3538_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_3539_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_3540_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd3: list_VEBT_VEBT,Ve3: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd3),Ve3),Vf2) = none_nat ).

% vebt_pred.simps(5)
tff(fact_3541_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_mint(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A2)
       => ( ( pp(B2)
           => ( vEBT_vebt_mint(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = some_nat(one_one_nat) ) )
          & ( ~ pp(B2)
           => ( vEBT_vebt_mint(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_3542_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: bool,A2: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_maxt(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = some_nat(one_one_nat) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_maxt(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_maxt(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_3543_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Vj2) = none_nat ).

% vebt_pred.simps(6)
tff(fact_3544_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),Uw),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A2)
       => ( vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A2),Uw),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_3545_linear__plus__1__le__power,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_3546_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts)))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts),Tsi),vEBT_vebt_maxti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bt(I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_3547_vebt__mintilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts)))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts),Tsi),vEBT_vebt_minti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bu(I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_3548_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3549_less__shift,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(X2)),some_nat(Y3))) ) ).

% less_shift
tff(fact_3550_greater__shift,axiom,
    ! [Y3: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(X2)),some_nat(Y3))) ) ).

% greater_shift
tff(fact_3551_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_vebt_maxti(T)),one_one_nat)) ).

% TBOUND_vebt_maxti
tff(fact_3552_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_vebt_minti(T)),one_one_nat)) ).

% TBOUND_vebt_minti
tff(fact_3553_max_Oidem,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),A2) = A2 ).

% max.idem
tff(fact_3554_max_Oidem,axiom,
    ! [A2: extended_enat] : ord_ma741700101516333627d_enat(A2,A2) = A2 ).

% max.idem
tff(fact_3555_max_Oidem,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A2),A2) = A2 ).

% max.idem
tff(fact_3556_max_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ).

% max.left_idem
tff(fact_3557_max_Oleft__idem,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ord_ma741700101516333627d_enat(A2,ord_ma741700101516333627d_enat(A2,B2)) = ord_ma741700101516333627d_enat(A2,B2) ).

% max.left_idem
tff(fact_3558_max_Oleft__idem,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) ).

% max.left_idem
tff(fact_3559_max_Oright__idem,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ).

% max.right_idem
tff(fact_3560_max_Oright__idem,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ord_ma741700101516333627d_enat(ord_ma741700101516333627d_enat(A2,B2),B2) = ord_ma741700101516333627d_enat(A2,B2) ).

% max.right_idem
tff(fact_3561_max_Oright__idem,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) ).

% max.right_idem
tff(fact_3562_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_minti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bv(T),Ti)) ).

% vebt_minti_h
tff(fact_3563_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_maxti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bw(T),Ti)) ).

% vebt_maxti_h
tff(fact_3564_max_Oabsorb1,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
     => ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ).

% max.absorb1
tff(fact_3565_max_Oabsorb1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_3566_max_Oabsorb1,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_3567_max_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_3568_max_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_3569_max_Oabsorb2,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),B2))
     => ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) ) ).

% max.absorb2
tff(fact_3570_max_Oabsorb2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_3571_max_Oabsorb2,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( aa_num_num(ord_max_num(A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_3572_max_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_3573_max_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_3574_max_Obounded__iff,axiom,
    ! [B2: extended_enat,C2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B2,C2)),A2))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
        & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),A2)) ) ) ).

% max.bounded_iff
tff(fact_3575_max_Obounded__iff,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2)) ) ) ).

% max.bounded_iff
tff(fact_3576_max_Obounded__iff,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2)) ) ) ).

% max.bounded_iff
tff(fact_3577_max_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2)) ) ) ).

% max.bounded_iff
tff(fact_3578_max_Obounded__iff,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2)) ) ) ).

% max.bounded_iff
tff(fact_3579_max__less__iff__conj,axiom,
    ! [X2: extended_enat,Y3: extended_enat,Z: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(X2,Y3)),Z))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X2),Z))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Y3),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3580_max__less__iff__conj,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y3)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3581_max__less__iff__conj,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(X2),Y3)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3582_max__less__iff__conj,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(X2),Y3)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3583_max__less__iff__conj,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3584_max__less__iff__conj,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z)) ) ) ).

% max_less_iff_conj
tff(fact_3585_max_Oabsorb4,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A2),B2))
     => ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) ) ).

% max.absorb4
tff(fact_3586_max_Oabsorb4,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3587_max_Oabsorb4,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3588_max_Oabsorb4,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( aa_num_num(ord_max_num(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3589_max_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3590_max_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3591_max_Oabsorb3,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B2),A2))
     => ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ).

% max.absorb3
tff(fact_3592_max_Oabsorb3,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_3593_max_Oabsorb3,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_3594_max_Oabsorb3,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_3595_max_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_3596_max_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_3597_max_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)) ).

% max.assoc
tff(fact_3598_max_Oassoc,axiom,
    ! [A2: extended_enat,B2: extended_enat,C2: extended_enat] : ord_ma741700101516333627d_enat(ord_ma741700101516333627d_enat(A2,B2),C2) = ord_ma741700101516333627d_enat(A2,ord_ma741700101516333627d_enat(B2,C2)) ).

% max.assoc
tff(fact_3599_max_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)) ).

% max.assoc
tff(fact_3600_max_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),A2) ).

% max.commute
tff(fact_3601_max_Ocommute,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ord_ma741700101516333627d_enat(A2,B2) = ord_ma741700101516333627d_enat(B2,A2) ).

% max.commute
tff(fact_3602_max_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = aa_int_int(aa_int_fun_int_int(ord_max_int,B2),A2) ).

% max.commute
tff(fact_3603_max_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)) ).

% max.left_commute
tff(fact_3604_max_Oleft__commute,axiom,
    ! [B2: extended_enat,A2: extended_enat,C2: extended_enat] : ord_ma741700101516333627d_enat(B2,ord_ma741700101516333627d_enat(A2,C2)) = ord_ma741700101516333627d_enat(A2,ord_ma741700101516333627d_enat(B2,C2)) ).

% max.left_commute
tff(fact_3605_max_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,B2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),C2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)) ).

% max.left_commute
tff(fact_3606_max_Omono,axiom,
    ! [C2: extended_enat,A2: extended_enat,D2: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),A2))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(D2),B2))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(C2,D2)),ord_ma741700101516333627d_enat(A2,B2))) ) ) ).

% max.mono
tff(fact_3607_max_Omono,axiom,
    ! [C2: rat,A2: rat,D2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C2),D2)),aa_rat_rat(ord_max_rat(A2),B2))) ) ) ).

% max.mono
tff(fact_3608_max_Omono,axiom,
    ! [C2: num,A2: num,D2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(C2),D2)),aa_num_num(ord_max_num(A2),B2))) ) ) ).

% max.mono
tff(fact_3609_max_Omono,axiom,
    ! [C2: nat,A2: nat,D2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ) ) ).

% max.mono
tff(fact_3610_max_Omono,axiom,
    ! [C2: int,A2: int,D2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C2),D2)),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ) ) ).

% max.mono
tff(fact_3611_max_OorderE,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
     => ( A2 = ord_ma741700101516333627d_enat(A2,B2) ) ) ).

% max.orderE
tff(fact_3612_max_OorderE,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( A2 = aa_rat_rat(ord_max_rat(A2),B2) ) ) ).

% max.orderE
tff(fact_3613_max_OorderE,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( A2 = aa_num_num(ord_max_num(A2),B2) ) ) ).

% max.orderE
tff(fact_3614_max_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ) ) ).

% max.orderE
tff(fact_3615_max_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) ) ) ).

% max.orderE
tff(fact_3616_max_OorderI,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( A2 = ord_ma741700101516333627d_enat(A2,B2) )
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2)) ) ).

% max.orderI
tff(fact_3617_max_OorderI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(ord_max_rat(A2),B2) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ).

% max.orderI
tff(fact_3618_max_OorderI,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = aa_num_num(ord_max_num(A2),B2) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ).

% max.orderI
tff(fact_3619_max_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ).

% max.orderI
tff(fact_3620_max_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% max.orderI
tff(fact_3621_max_OboundedE,axiom,
    ! [B2: extended_enat,C2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B2,C2)),A2))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),A2)) ) ) ).

% max.boundedE
tff(fact_3622_max_OboundedE,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2)) ) ) ).

% max.boundedE
tff(fact_3623_max_OboundedE,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A2))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2)) ) ) ).

% max.boundedE
tff(fact_3624_max_OboundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2)) ) ) ).

% max.boundedE
tff(fact_3625_max_OboundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2)) ) ) ).

% max.boundedE
tff(fact_3626_max_OboundedI,axiom,
    ! [B2: extended_enat,A2: extended_enat,C2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),A2))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B2,C2)),A2)) ) ) ).

% max.boundedI
tff(fact_3627_max_OboundedI,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2)) ) ) ).

% max.boundedI
tff(fact_3628_max_OboundedI,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A2)) ) ) ).

% max.boundedI
tff(fact_3629_max_OboundedI,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2)) ) ) ).

% max.boundedI
tff(fact_3630_max_OboundedI,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A2)) ) ) ).

% max.boundedI
tff(fact_3631_max_Oorder__iff,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
    <=> ( A2 = ord_ma741700101516333627d_enat(A2,B2) ) ) ).

% max.order_iff
tff(fact_3632_max_Oorder__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
    <=> ( A2 = aa_rat_rat(ord_max_rat(A2),B2) ) ) ).

% max.order_iff
tff(fact_3633_max_Oorder__iff,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> ( A2 = aa_num_num(ord_max_num(A2),B2) ) ) ).

% max.order_iff
tff(fact_3634_max_Oorder__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ) ) ).

% max.order_iff
tff(fact_3635_max_Oorder__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) ) ) ).

% max.order_iff
tff(fact_3636_max_Ocobounded1,axiom,
    ! [A2: extended_enat,B2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),ord_ma741700101516333627d_enat(A2,B2))) ).

% max.cobounded1
tff(fact_3637_max_Ocobounded1,axiom,
    ! [A2: rat,B2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(ord_max_rat(A2),B2))) ).

% max.cobounded1
tff(fact_3638_max_Ocobounded1,axiom,
    ! [A2: num,B2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(ord_max_num(A2),B2))) ).

% max.cobounded1
tff(fact_3639_max_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ).

% max.cobounded1
tff(fact_3640_max_Ocobounded1,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ).

% max.cobounded1
tff(fact_3641_max_Ocobounded2,axiom,
    ! [B2: extended_enat,A2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),ord_ma741700101516333627d_enat(A2,B2))) ).

% max.cobounded2
tff(fact_3642_max_Ocobounded2,axiom,
    ! [B2: rat,A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),aa_rat_rat(ord_max_rat(A2),B2))) ).

% max.cobounded2
tff(fact_3643_max_Ocobounded2,axiom,
    ! [B2: num,A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),aa_num_num(ord_max_num(A2),B2))) ).

% max.cobounded2
tff(fact_3644_max_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ).

% max.cobounded2
tff(fact_3645_max_Ocobounded2,axiom,
    ! [B2: int,A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ).

% max.cobounded2
tff(fact_3646_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X2: extended_enat,Y3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),ord_ma741700101516333627d_enat(X2,Y3)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),X2))
        | pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y3)) ) ) ).

% le_max_iff_disj
tff(fact_3647_le__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(X2),Y3)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X2))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y3)) ) ) ).

% le_max_iff_disj
tff(fact_3648_le__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(ord_max_num(X2),Y3)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X2))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y3)) ) ) ).

% le_max_iff_disj
tff(fact_3649_le__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y3)) ) ) ).

% le_max_iff_disj
tff(fact_3650_le__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y3)) ) ) ).

% le_max_iff_disj
tff(fact_3651_max_Oabsorb__iff1,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B2),A2))
    <=> ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_3652_max_Oabsorb__iff1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
    <=> ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_3653_max_Oabsorb__iff1,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_3654_max_Oabsorb__iff1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_3655_max_Oabsorb__iff1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_3656_max_Oabsorb__iff2,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),B2))
    <=> ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_3657_max_Oabsorb__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
    <=> ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_3658_max_Oabsorb__iff2,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
    <=> ( aa_num_num(ord_max_num(A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_3659_max_Oabsorb__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_3660_max_Oabsorb__iff2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_3661_max_OcoboundedI1,axiom,
    ! [C2: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),A2))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),ord_ma741700101516333627d_enat(A2,B2))) ) ).

% max.coboundedI1
tff(fact_3662_max_OcoboundedI1,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A2),B2))) ) ).

% max.coboundedI1
tff(fact_3663_max_OcoboundedI1,axiom,
    ! [C2: num,A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),aa_num_num(ord_max_num(A2),B2))) ) ).

% max.coboundedI1
tff(fact_3664_max_OcoboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ) ).

% max.coboundedI1
tff(fact_3665_max_OcoboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ) ).

% max.coboundedI1
tff(fact_3666_max_OcoboundedI2,axiom,
    ! [C2: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),B2))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C2),ord_ma741700101516333627d_enat(A2,B2))) ) ).

% max.coboundedI2
tff(fact_3667_max_OcoboundedI2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A2),B2))) ) ).

% max.coboundedI2
tff(fact_3668_max_OcoboundedI2,axiom,
    ! [C2: num,B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),aa_num_num(ord_max_num(A2),B2))) ) ).

% max.coboundedI2
tff(fact_3669_max_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ) ).

% max.coboundedI2
tff(fact_3670_max_OcoboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ) ).

% max.coboundedI2
tff(fact_3671_max_Ostrict__coboundedI2,axiom,
    ! [C2: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C2),B2))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C2),ord_ma741700101516333627d_enat(A2,B2))) ) ).

% max.strict_coboundedI2
tff(fact_3672_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2))) ) ).

% max.strict_coboundedI2
tff(fact_3673_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A2),B2))) ) ).

% max.strict_coboundedI2
tff(fact_3674_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),B2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),aa_num_num(ord_max_num(A2),B2))) ) ).

% max.strict_coboundedI2
tff(fact_3675_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ) ).

% max.strict_coboundedI2
tff(fact_3676_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ) ).

% max.strict_coboundedI2
tff(fact_3677_max_Ostrict__coboundedI1,axiom,
    ! [C2: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C2),A2))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C2),ord_ma741700101516333627d_enat(A2,B2))) ) ).

% max.strict_coboundedI1
tff(fact_3678_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2))) ) ).

% max.strict_coboundedI1
tff(fact_3679_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A2),B2))) ) ).

% max.strict_coboundedI1
tff(fact_3680_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),aa_num_num(ord_max_num(A2),B2))) ) ).

% max.strict_coboundedI1
tff(fact_3681_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2))) ) ).

% max.strict_coboundedI1
tff(fact_3682_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2))) ) ).

% max.strict_coboundedI1
tff(fact_3683_max_Ostrict__order__iff,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B2),A2))
    <=> ( ( A2 = ord_ma741700101516333627d_enat(A2,B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_3684_max_Ostrict__order__iff,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
    <=> ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_3685_max_Ostrict__order__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
    <=> ( ( A2 = aa_rat_rat(ord_max_rat(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_3686_max_Ostrict__order__iff,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
    <=> ( ( A2 = aa_num_num(ord_max_num(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_3687_max_Ostrict__order__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_3688_max_Ostrict__order__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_3689_max_Ostrict__boundedE,axiom,
    ! [B2: extended_enat,C2: extended_enat,A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(B2,C2)),A2))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B2),A2))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C2),A2)) ) ) ).

% max.strict_boundedE
tff(fact_3690_max_Ostrict__boundedE,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B2),C2)),A2))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A2)) ) ) ).

% max.strict_boundedE
tff(fact_3691_max_Ostrict__boundedE,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A2)) ) ) ).

% max.strict_boundedE
tff(fact_3692_max_Ostrict__boundedE,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(B2),C2)),A2))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A2)) ) ) ).

% max.strict_boundedE
tff(fact_3693_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A2)) ) ) ).

% max.strict_boundedE
tff(fact_3694_max_Ostrict__boundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C2)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A2)) ) ) ).

% max.strict_boundedE
tff(fact_3695_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X2: extended_enat,Y3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),ord_ma741700101516333627d_enat(X2,Y3)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),X2))
        | pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),Y3)) ) ) ).

% less_max_iff_disj
tff(fact_3696_less__max__iff__disj,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y3)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y3)) ) ) ).

% less_max_iff_disj
tff(fact_3697_less__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(ord_max_rat(X2),Y3)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y3)) ) ) ).

% less_max_iff_disj
tff(fact_3698_less__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(ord_max_num(X2),Y3)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y3)) ) ) ).

% less_max_iff_disj
tff(fact_3699_less__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y3)) ) ) ).

% less_max_iff_disj
tff(fact_3700_less__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y3)) ) ) ).

% less_max_iff_disj
tff(fact_3701_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( ! [A3: real,B3: real,C3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),C3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),B3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),C3)) ) ) ) )
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
               => ? [D5: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
                    & ! [A3: real,B3: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),X4))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B3))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D5)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A2),B2)) ) ) ) ).

% Bolzano
tff(fact_3702_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_maxti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bw(T),Ti)),one_one_nat)) ).

% vebt_maxti_hT
tff(fact_3703_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_minti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bv(T),Ti)),one_one_nat)) ).

% vebt_minti_hT
tff(fact_3704_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,X2),Xa) = Y3 )
     => ( ( ? [Uu2: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)
           => ( ? [N3: nat] : Xa = aa_nat_nat(suc,N3)
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)
                 => ( Y3 != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y3 = one_one_nat ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_3705_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,X2),Xa) = Y3 )
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A3: bool,Uw2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
             => ( ? [Va3: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va3))
               => ( Y3 != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y3 = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_3706_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_3707_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_3708_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: bool,Uu: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_succ(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B2),zero_zero_nat) = some_nat(one_one_nat) ) )
      & ( ~ pp(B2)
       => ( vEBT_vebt_succ(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B2),zero_zero_nat) = none_nat ) ) ) ).

% vebt_succ.simps(1)
tff(fact_3709_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,Va: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_3710_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_3711_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_bool_VEBT_VEBT(vEBT_Leaf(Uv),Uw)),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_3712_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_3713_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: bool,B2: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B2)),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_3714_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_3715_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),Uw)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_3716_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd3: list_VEBT_VEBT,Ve3: vEBT_VEBT,Vf2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd3),Ve3)),Vf2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_3717_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd3: vEBT_VEBT,Ve3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd3)),Ve3) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_3718_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_3719_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_3720_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% pred_bound_height'
tff(fact_3721_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% succ'_bound_height
tff(fact_3722_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X2),Y3))
    <=> ( pp(aa_set_nat_bool(member_nat(Y3),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),X3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X3)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_3723_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ'
tff(fact_3724_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ'
tff(fact_3725_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_vebt_succ(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv),Uw),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3726_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_3727_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd3: vEBT_VEBT,Ve3: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd3),Ve3) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3728_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Vi2) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3729_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_3730_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_3731_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_3732_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X2: nat,Y3: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Y3))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Y3)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_3733_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb))),aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,none_nat),none_nat)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),N))))),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Xc),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,N,Xa),Xb))))),H))
     => pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,N,TreeList),Summary)),Xc)),H)) ) ).

% heaphelp
tff(fact_3734_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xa),Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb))),aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,none_P5556105721700978146at_nat),none_P5556105721700978146at_nat)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),N))))),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Xc),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,N,Xa),Xb))))),H))
     => pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,N,TreeList),Summary)),Xc)),H)) ) ).

% heaphelp
tff(fact_3735_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P,N3))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_3736_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa))
     => ( ! [Uu2: bool,Uv2: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2)
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3737_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa))
      <=> pp(Y3) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => pp(Y3) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => pp(Y3) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2)
               => ( pp(Y3)
                <=> ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( pp(Y3)
                  <=> ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( pp(Y3)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3738_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)),X2)) ).

% buildup_nothing_in_min_max
tff(fact_3739_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(P),Q)),produc7507926704131184380et_nat(H,bot_bot_set_nat)))
    <=> ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H,bot_bot_set_nat)))
        & pp(aa_Pro2991721470636411351t_bool(rep_assn(Q),produc7507926704131184380et_nat(H,bot_bot_set_nat))) ) ) ).

% mod_h_bot_iff(5)
tff(fact_3740_mod__pure__star__dist,axiom,
    ! [P: assn,B2: bool,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2))),H))
    <=> ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H))
        & pp(B2) ) ) ).

% mod_pure_star_dist
tff(fact_3741_mod__h__bot__iff_I4_J,axiom,
    ! [Q3: array_VEBT_VEBTi,Y3: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : ~ pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Q3),Y3)),produc7507926704131184380et_nat(H,bot_bot_set_nat))) ).

% mod_h_bot_iff(4)
tff(fact_3742_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: bool] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(Q),aa_bool_assn(pure_assn,B2))))
    <=> ( ! [H3: produc3658429121746597890et_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H3))
           => pp(B2) )
        & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ).

% ent_pure_post_iff
tff(fact_3743_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: bool] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_bool_assn(pure_assn,B2)))
    <=> ( ! [H3: produc3658429121746597890et_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H3))
           => pp(B2) )
        & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),one_one_assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_3744_mod__starE,axiom,
    ! [A2: assn,B2: assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(A2),B2)),H))
     => ~ ( ? [X_12: produc3658429121746597890et_nat] : pp(aa_Pro2991721470636411351t_bool(rep_assn(A2),X_12))
         => ! [H_2: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(rep_assn(B2),H_2)) ) ) ).

% mod_starE
tff(fact_3745_mod__starD,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(A),B)),H))
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(rep_assn(A),H1))
          & pp(aa_Pro2991721470636411351t_bool(rep_assn(B),H22)) ) ) ).

% mod_starD
tff(fact_3746_ent__fwd,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
       => pp(aa_Pro2991721470636411351t_bool(rep_assn(Q),H)) ) ) ).

% ent_fwd
tff(fact_3747_entailsD,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H))
       => pp(aa_Pro2991721470636411351t_bool(rep_assn(Q),H)) ) ) ).

% entailsD
tff(fact_3748_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H2))
         => pp(aa_Pro2991721470636411351t_bool(rep_assn(Q),H2)) )
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ).

% entailsI
tff(fact_3749_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
    <=> ! [H3: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H3))
         => pp(aa_Pro2991721470636411351t_bool(rep_assn(Q),H3)) ) ) ).

% entails_def
tff(fact_3750_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N))
          & pp(aa_nat_bool(P,M7)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less
tff(fact_3751_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N))
         => pp(aa_nat_bool(P,M7)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less
tff(fact_3752_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : pp(aa_Pro2991721470636411351t_bool(rep_assn(one_one_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat))) ).

% mod_emp_simp
tff(fact_3753_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
           => ( ( pp(aa_real_bool(P,X6))
                & pp(aa_real_bool(Q,X6)) )
            <=> ( pp(aa_real_bool(P3,X6))
                & pp(aa_real_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3754_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
           => ( ( pp(aa_rat_bool(P,X6))
                & pp(aa_rat_bool(Q,X6)) )
            <=> ( pp(aa_rat_bool(P3,X6))
                & pp(aa_rat_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3755_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
           => ( ( pp(aa_num_bool(P,X6))
                & pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P3,X6))
                & pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3756_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
           => ( ( pp(aa_nat_bool(P,X6))
                & pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P3,X6))
                & pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3757_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P3,X6))
                & pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_3758_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
           => ( ( pp(aa_real_bool(P,X6))
                | pp(aa_real_bool(Q,X6)) )
            <=> ( pp(aa_real_bool(P3,X6))
                | pp(aa_real_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3759_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
           => ( ( pp(aa_rat_bool(P,X6))
                | pp(aa_rat_bool(Q,X6)) )
            <=> ( pp(aa_rat_bool(P3,X6))
                | pp(aa_rat_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3760_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
           => ( ( pp(aa_num_bool(P,X6))
                | pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P3,X6))
                | pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3761_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
           => ( ( pp(aa_nat_bool(P,X6))
                | pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P3,X6))
                | pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3762_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P3,X6))
                | pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_3763_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_3764_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_3765_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_3766_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_3767_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_3768_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_3769_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_3770_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_3771_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_3772_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_3773_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),T)) ) ).

% pinf(5)
tff(fact_3774_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),T)) ) ).

% pinf(5)
tff(fact_3775_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),T)) ) ).

% pinf(5)
tff(fact_3776_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),T)) ) ).

% pinf(5)
tff(fact_3777_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T)) ) ).

% pinf(5)
tff(fact_3778_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X6)) ) ).

% pinf(7)
tff(fact_3779_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X6)) ) ).

% pinf(7)
tff(fact_3780_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X6)) ) ).

% pinf(7)
tff(fact_3781_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X6)) ) ).

% pinf(7)
tff(fact_3782_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6)) ) ).

% pinf(7)
tff(fact_3783_minf_I1_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
           => ( ( pp(aa_real_bool(P,X6))
                & pp(aa_real_bool(Q,X6)) )
            <=> ( pp(aa_real_bool(P3,X6))
                & pp(aa_real_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_3784_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
           => ( ( pp(aa_rat_bool(P,X6))
                & pp(aa_rat_bool(Q,X6)) )
            <=> ( pp(aa_rat_bool(P3,X6))
                & pp(aa_rat_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_3785_minf_I1_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
           => ( ( pp(aa_num_bool(P,X6))
                & pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P3,X6))
                & pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_3786_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
           => ( ( pp(aa_nat_bool(P,X6))
                & pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P3,X6))
                & pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_3787_minf_I1_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P3,X6))
                & pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_3788_minf_I2_J,axiom,
    ! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
           => ( ( pp(aa_real_bool(P,X6))
                | pp(aa_real_bool(Q,X6)) )
            <=> ( pp(aa_real_bool(P3,X6))
                | pp(aa_real_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_3789_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
           => ( ( pp(aa_rat_bool(P,X6))
                | pp(aa_rat_bool(Q,X6)) )
            <=> ( pp(aa_rat_bool(P3,X6))
                | pp(aa_rat_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_3790_minf_I2_J,axiom,
    ! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
           => ( ( pp(aa_num_bool(P,X6))
                | pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P3,X6))
                | pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_3791_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
           => ( ( pp(aa_nat_bool(P,X6))
                | pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P3,X6))
                | pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_3792_minf_I2_J,axiom,
    ! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P3,X6))
                | pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_3793_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_3794_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_3795_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_3796_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_3797_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_3798_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_3799_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_3800_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_3801_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_3802_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_3803_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),T)) ) ).

% minf(5)
tff(fact_3804_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),T)) ) ).

% minf(5)
tff(fact_3805_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),T)) ) ).

% minf(5)
tff(fact_3806_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),T)) ) ).

% minf(5)
tff(fact_3807_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T)) ) ).

% minf(5)
tff(fact_3808_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X6)) ) ).

% minf(7)
tff(fact_3809_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X6)) ) ).

% minf(7)
tff(fact_3810_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X6)) ) ).

% minf(7)
tff(fact_3811_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X6)) ) ).

% minf(7)
tff(fact_3812_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6)) ) ).

% minf(7)
tff(fact_3813_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3814_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb)),X2))
    <=> ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3815_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X6),T)) ) ).

% pinf(6)
tff(fact_3816_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X6),T)) ) ).

% pinf(6)
tff(fact_3817_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X6),T)) ) ).

% pinf(6)
tff(fact_3818_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),T)) ) ).

% pinf(6)
tff(fact_3819_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T)) ) ).

% pinf(6)
tff(fact_3820_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X6)) ) ).

% pinf(8)
tff(fact_3821_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X6)) ) ).

% pinf(8)
tff(fact_3822_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X6))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X6)) ) ).

% pinf(8)
tff(fact_3823_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X6)) ) ).

% pinf(8)
tff(fact_3824_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6)) ) ).

% pinf(8)
tff(fact_3825_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X6),T)) ) ).

% minf(6)
tff(fact_3826_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X6),T)) ) ).

% minf(6)
tff(fact_3827_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X6),T)) ) ).

% minf(6)
tff(fact_3828_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),T)) ) ).

% minf(6)
tff(fact_3829_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T)) ) ).

% minf(6)
tff(fact_3830_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X6)) ) ).

% minf(8)
tff(fact_3831_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X6)) ) ).

% minf(8)
tff(fact_3832_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X6)) ) ).

% minf(8)
tff(fact_3833_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X6)) ) ).

% minf(8)
tff(fact_3834_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6)) ) ).

% minf(8)
tff(fact_3835_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D: real,Q: fun_real_bool] :
      ( ! [X4: real,K2: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D)))) )
     => ( ! [X4: real,K2: real] :
            ( pp(aa_real_bool(Q,X4))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D)))) )
       => ! [X6: real,K5: real] :
            ( ( pp(aa_real_bool(P,X6))
              & pp(aa_real_bool(Q,X6)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))))
              & pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3836_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D: rat,Q: fun_rat_bool] :
      ( ! [X4: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X4))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D)))) )
     => ( ! [X4: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X4))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D)))) )
       => ! [X6: rat,K5: rat] :
            ( ( pp(aa_rat_bool(P,X6))
              & pp(aa_rat_bool(Q,X6)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X6),aa_rat_rat(times_times_rat(K5),D))))
              & pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X6),aa_rat_rat(times_times_rat(K5),D)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3837_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D: int,Q: fun_int_bool] :
      ( ! [X4: int,K2: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(Q,X4))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ! [X6: int,K5: int] :
            ( ( pp(aa_int_bool(P,X6))
              & pp(aa_int_bool(Q,X6)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3838_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D: real,Q: fun_real_bool] :
      ( ! [X4: real,K2: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D)))) )
     => ( ! [X4: real,K2: real] :
            ( pp(aa_real_bool(Q,X4))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D)))) )
       => ! [X6: real,K5: real] :
            ( ( pp(aa_real_bool(P,X6))
              | pp(aa_real_bool(Q,X6)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))))
              | pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3839_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D: rat,Q: fun_rat_bool] :
      ( ! [X4: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X4))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D)))) )
     => ( ! [X4: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X4))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D)))) )
       => ! [X6: rat,K5: rat] :
            ( ( pp(aa_rat_bool(P,X6))
              | pp(aa_rat_bool(Q,X6)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X6),aa_rat_rat(times_times_rat(K5),D))))
              | pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X6),aa_rat_rat(times_times_rat(K5),D)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3840_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D: int,Q: fun_int_bool] :
      ( ! [X4: int,K2: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(Q,X4))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ! [X6: int,K5: int] :
            ( ( pp(aa_int_bool(P,X6))
              | pp(aa_int_bool(Q,X6)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3841_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( X2 != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va3: nat] : X2 != aa_nat_nat(suc,aa_nat_nat(suc,Va3)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_3842_conj__le__cong,axiom,
    ! [X2: int,X5: int,P: bool,P3: bool] :
      ( ( X2 = X5 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
            & pp(P3) ) ) ) ) ).

% conj_le_cong
tff(fact_3843_imp__le__cong,axiom,
    ! [X2: int,X5: int,P: bool,P3: bool] :
      ( ( X2 = X5 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
           => pp(P3) ) ) ) ) ).

% imp_le_cong
tff(fact_3844_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = one_one_real ).

% VEBT_internal.cnt.simps(1)
tff(fact_3845_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = one_one_nat ).

% VEBT_internal.cnt'.simps(1)
tff(fact_3846_plusinfinity,axiom,
    ! [D2: int,P3: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P3,X4))
          <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2)))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P3,X4)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P3,X_13))
           => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).

% plusinfinity
tff(fact_3847_minusinfinity,axiom,
    ! [D2: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P1,X4))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2)))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P1,X4)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P1,X_13))
           => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).

% minusinfinity
tff(fact_3848_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd3: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),Vd3)),X2))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3849_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList),Vc)),X2))
    <=> ( ( X2 = Mi )
        | ( X2 = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3850_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% double_not_eq_Suc_double
tff(fact_3851_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_double_not_eq_double
tff(fact_3852_incr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D2))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X6: int] :
              ( pp(aa_int_bool(P,X6))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3853_decr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D2))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X6: int] :
              ( pp(aa_int_bool(P,X6))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3854_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_3855_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_3856_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_3857_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_3858_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2)
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3859_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X2))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,Tree),X2))
          | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,Tree),X2)) ) ) ) ).

% member_valid_both_member_options
tff(fact_3860_atLeastatMost__subset__iff,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int,D2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A2,B2),set_or370866239135849197et_int(C2,D2))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        | ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A2))
          & pp(aa_set_int_bool(ord_less_eq_set_int(B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3861_atLeastatMost__subset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or633870826150836451st_rat(C2,D2)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3862_atLeastatMost__subset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A2,B2)),set_or7049704709247886629st_num(C2,D2)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3863_atLeastatMost__subset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_or1269000886237332187st_nat(A2,B2)),set_or1269000886237332187st_nat(C2,D2)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3864_atLeastatMost__subset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A2,B2)),set_or1266510415728281911st_int(C2,D2)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3865_atLeastatMost__subset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A2,B2)),set_or189985376899183464nteger(C2,D2)))
    <=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3866_atLeastatMost__subset__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A2,B2)),set_or1222579329274155063t_real(C2,D2)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),D2)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3867_atLeastatMost__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( set_or633870826150836451st_rat(A2,B2) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3868_atLeastatMost__empty,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( set_or7049704709247886629st_num(A2,B2) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3869_atLeastatMost__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( set_or1269000886237332187st_nat(A2,B2) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3870_atLeastatMost__empty,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( set_or1266510415728281911st_int(A2,B2) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3871_atLeastatMost__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( set_or189985376899183464nteger(A2,B2) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_3872_atLeastatMost__empty,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( set_or1222579329274155063t_real(A2,B2) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3873_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A2,B2) )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3874_atLeastatMost__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A2,B2) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3875_atLeastatMost__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A2,B2) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3876_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A2,B2) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3877_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A2,B2) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3878_atLeastatMost__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or189985376899183464nteger(A2,B2) )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3879_atLeastatMost__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A2,B2) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3880_atLeastatMost__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( set_or370866239135849197et_int(A2,B2) = bot_bot_set_set_int )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3881_atLeastatMost__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( set_or633870826150836451st_rat(A2,B2) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3882_atLeastatMost__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( set_or7049704709247886629st_num(A2,B2) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3883_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( set_or1269000886237332187st_nat(A2,B2) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3884_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( set_or1266510415728281911st_int(A2,B2) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3885_atLeastatMost__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( set_or189985376899183464nteger(A2,B2) = bot_bo3990330152332043303nteger )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3886_atLeastatMost__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( set_or1222579329274155063t_real(A2,B2) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3887_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,T),X2))
        | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,T),X2)) ) ) ).

% both_member_options_def
tff(fact_3888_nat__approx__posE,axiom,
    ! [E: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
     => ~ ! [N3: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N3)))),E)) ) ).

% nat_approx_posE
tff(fact_3889_nat__approx__posE,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ~ ! [N3: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))),E)) ) ).

% nat_approx_posE
tff(fact_3890_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)),X2)) ).

% buildup_nothing_in_leaf
tff(fact_3891_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L3: set_int,H4: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
          & ~ pp(aa_set_int_bool(ord_less_eq_set_int(L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3892_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H4: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
          & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3893_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H4: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
          & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3894_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H4: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
          & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3895_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H4: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
          & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3896_Icc__eq__Icc,axiom,
    ! [L: code_integer,H: code_integer,L3: code_integer,H4: code_integer] :
      ( ( set_or189985376899183464nteger(L,H) = set_or189985376899183464nteger(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),H))
          & ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3897_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H4: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L3,H4) )
    <=> ( ( ( L = L3 )
          & ( H = H4 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),H4)) ) ) ) ).

% Icc_eq_Icc
tff(fact_3898_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I))
        & pp(aa_set_int_bool(ord_less_eq_set_int(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3899_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or633870826150836451st_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3900_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or7049704709247886629st_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3901_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),set_or1269000886237332187st_nat(L,U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3902_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3903_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),set_or189985376899183464nteger(L,U)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),I))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3904_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or1222579329274155063t_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3905_aset_I2_J,axiom,
    ! [D: int,A: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ) ) ).

% aset(2)
tff(fact_3906_aset_I1_J,axiom,
    ! [D: int,A: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ) ) ).

% aset(1)
tff(fact_3907_bset_I2_J,axiom,
    ! [D: int,B: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ) ).

% bset(2)
tff(fact_3908_bset_I1_J,axiom,
    ! [D: int,B: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ) ).

% bset(1)
tff(fact_3909_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_3910_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X2: nat,M8: nat] :
      ( pp(aa_nat_bool(P,X2))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M8)) )
       => ~ ! [M5: nat] :
              ( pp(aa_nat_bool(P,M5))
             => ~ ! [X6: nat] :
                    ( pp(aa_nat_bool(P,X6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_3911_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X2 != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_3912_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2))
    <=> ( ( ( X2 = zero_zero_nat )
         => pp(A2) )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => pp(B2) )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3913_aset_I7_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ).

% aset(7)
tff(fact_3914_aset_I5_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(T),A))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T)) ) ) ) ) ).

% aset(5)
tff(fact_3915_aset_I4_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(T),A))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X6 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D) != T ) ) ) ) ) ).

% aset(4)
tff(fact_3916_aset_I3_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X6 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D) = T ) ) ) ) ) ).

% aset(3)
tff(fact_3917_bset_I7_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(T),B))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ).

% bset(7)
tff(fact_3918_bset_I5_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T)) ) ) ) ).

% bset(5)
tff(fact_3919_bset_I4_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(T),B))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X6 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D) != T ) ) ) ) ) ).

% bset(4)
tff(fact_3920_bset_I3_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X6 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D) = T ) ) ) ) ) ).

% bset(3)
tff(fact_3921_periodic__finite__ex,axiom,
    ! [D2: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2)))) )
       => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D2)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3922_bset_I6_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T)) ) ) ) ).

% bset(6)
tff(fact_3923_bset_I8_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ).

% bset(8)
tff(fact_3924_aset_I6_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A))
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A))
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T)) ) ) ) ) ).

% aset(6)
tff(fact_3925_aset_I8_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ).

% aset(8)
tff(fact_3926_cpmi,axiom,
    ! [D: int,P: fun_int_bool,P3: fun_int_bool,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z4))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(P3,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
                 => ! [Xb2: int] :
                      ( pp(aa_set_int_bool(member_int(Xb2),B))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa_int_bool(P3,X4))
              <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
           => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
                    & pp(aa_int_bool(P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),B))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3927_cppi,axiom,
    ! [D: int,P: fun_int_bool,P3: fun_int_bool,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X4))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(P3,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D)))
                 => ! [Xb2: int] :
                      ( pp(aa_set_int_bool(member_int(Xb2),A))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa_int_bool(P3,X4))
              <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
           => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
                    & pp(aa_int_bool(P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),A))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3928_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(P,N3))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) ) ) ).

% exists_least_lemma
tff(fact_3929_real__arch__simple,axiom,
    ! [X2: real] :
    ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(semiri5074537144036343181t_real,N3))) ).

% real_arch_simple
tff(fact_3930_real__arch__simple,axiom,
    ! [X2: rat] :
    ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_nat_rat(semiri681578069525770553at_rat,N3))) ).

% real_arch_simple
tff(fact_3931_reals__Archimedean2,axiom,
    ! [X2: rat] :
    ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_nat_rat(semiri681578069525770553at_rat,N3))) ).

% reals_Archimedean2
tff(fact_3932_reals__Archimedean2,axiom,
    ! [X2: real] :
    ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(semiri5074537144036343181t_real,N3))) ).

% reals_Archimedean2
tff(fact_3933_ex__less__of__nat__mult,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N3)),X2))) ) ).

% ex_less_of_nat_mult
tff(fact_3934_ex__less__of__nat__mult,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),X2))) ) ).

% ex_less_of_nat_mult
tff(fact_3935_atLeastatMost__psubset__iff,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int,D2: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A2,B2),set_or370866239135849197et_int(C2,D2))
    <=> ( ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
          | ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A2))
            & pp(aa_set_int_bool(ord_less_eq_set_int(B2),D2))
            & ( pp(aa_set_int_bool(ord_less_set_int(C2),A2))
              | pp(aa_set_int_bool(ord_less_set_int(B2),D2)) ) ) )
        & pp(aa_set_int_bool(ord_less_eq_set_int(C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3936_atLeastatMost__psubset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or633870826150836451st_rat(C2,D2)))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),D2))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A2))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),D2)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3937_atLeastatMost__psubset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A2,B2)),set_or7049704709247886629st_num(C2,D2)))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),D2))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A2))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),D2)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3938_atLeastatMost__psubset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(set_or1269000886237332187st_nat(A2,B2)),set_or1269000886237332187st_nat(C2,D2)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D2))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A2))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),D2)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3939_atLeastatMost__psubset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A2,B2)),set_or1266510415728281911st_int(C2,D2)))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D2))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A2))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),D2)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3940_atLeastatMost__psubset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A2,B2)),set_or189985376899183464nteger(C2,D2)))
    <=> ( ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
          | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),A2))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),D2))
            & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),A2))
              | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),D2)) ) ) )
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3941_atLeastatMost__psubset__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A2,B2)),set_or1222579329274155063t_real(C2,D2)))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),D2))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A2))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),D2)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3942_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList),S2)),X2))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3943_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ( ( ( Xa = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => pp(B3) )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3944_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => pp(B3) )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3945_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa))
      <=> pp(Y3) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ( pp(Y3)
            <=> ~ ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
           => pp(Y3) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3)
               => ( pp(Y3)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3946_htt__def,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C2,Q),T))
    <=> ( hoare_1429296392585015714_VEBTi(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H3,As)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(C2,H3)),T)) ) ) ) ).

% htt_def
tff(fact_3947_htt__def,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(P,C2,Q),T))
    <=> ( hoare_2774773592644289642e_bool(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H3,As)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(C2,H3)),T)) ) ) ) ).

% htt_def
tff(fact_3948_htt__def,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(P,C2,Q),T))
    <=> ( hoare_7629718768684598413on_nat(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H3,As)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(C2,H3)),T)) ) ) ) ).

% htt_def
tff(fact_3949_htt__def,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(P,C2,Q),T))
    <=> ( hoare_3067605981109127869le_nat(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H3,As)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(C2,H3)),T)) ) ) ) ).

% htt_def
tff(fact_3950_httI,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H2,As2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(C2,H2)),T)) )
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C2,Q),T)) ) ) ).

% httI
tff(fact_3951_httI,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( hoare_2774773592644289642e_bool(P,C2,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H2,As2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(C2,H2)),T)) )
       => pp(aa_nat_bool(time_htt_bool(P,C2,Q),T)) ) ) ).

% httI
tff(fact_3952_httI,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H2,As2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(C2,H2)),T)) )
       => pp(aa_nat_bool(time_htt_option_nat(P,C2,Q),T)) ) ) ).

% httI
tff(fact_3953_httI,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),produc7507926704131184380et_nat(H2,As2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(C2,H2)),T)) )
       => pp(aa_nat_bool(time_htt_nat(P,C2,Q),T)) ) ) ).

% httI
tff(fact_3954_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ) ) ).

% int_ops(6)
tff(fact_3955_max__bot2,axiom,
    ! [X2: set_nat] : ord_max_set_nat(X2,bot_bot_set_nat) = X2 ).

% max_bot2
tff(fact_3956_max__bot2,axiom,
    ! [X2: assn] : ord_max_assn(X2,bot_bot_assn) = X2 ).

% max_bot2
tff(fact_3957_max__bot2,axiom,
    ! [X2: set_int] : ord_max_set_int(X2,bot_bot_set_int) = X2 ).

% max_bot2
tff(fact_3958_max__bot2,axiom,
    ! [X2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),bot_bot_nat) = X2 ).

% max_bot2
tff(fact_3959_max__bot2,axiom,
    ! [X2: extended_enat] : ord_ma741700101516333627d_enat(X2,bot_bo4199563552545308370d_enat) = X2 ).

% max_bot2
tff(fact_3960_max__bot,axiom,
    ! [X2: set_nat] : ord_max_set_nat(bot_bot_set_nat,X2) = X2 ).

% max_bot
tff(fact_3961_max__bot,axiom,
    ! [X2: assn] : ord_max_assn(bot_bot_assn,X2) = X2 ).

% max_bot
tff(fact_3962_max__bot,axiom,
    ! [X2: set_int] : ord_max_set_int(bot_bot_set_int,X2) = X2 ).

% max_bot
tff(fact_3963_max__bot,axiom,
    ! [X2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X2) = X2 ).

% max_bot
tff(fact_3964_max__bot,axiom,
    ! [X2: extended_enat] : ord_ma741700101516333627d_enat(bot_bo4199563552545308370d_enat,X2) = X2 ).

% max_bot
tff(fact_3965_pos__mult__pos__ge,axiom,
    ! [X2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,N),X2))) ) ) ).

% pos_mult_pos_ge
tff(fact_3966_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% heigt_uplog_rel
tff(fact_3967_order__refl,axiom,
    ! [X2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X2),X2)) ).

% order_refl
tff(fact_3968_order__refl,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),X2)) ).

% order_refl
tff(fact_3969_order__refl,axiom,
    ! [X2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),X2)) ).

% order_refl
tff(fact_3970_order__refl,axiom,
    ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),X2)) ).

% order_refl
tff(fact_3971_order__refl,axiom,
    ! [X2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),X2)) ).

% order_refl
tff(fact_3972_dual__order_Orefl,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% dual_order.refl
tff(fact_3973_dual__order_Orefl,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),A2)) ).

% dual_order.refl
tff(fact_3974_dual__order_Orefl,axiom,
    ! [A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),A2)) ).

% dual_order.refl
tff(fact_3975_dual__order_Orefl,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A2)) ).

% dual_order.refl
tff(fact_3976_dual__order_Orefl,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A2)) ).

% dual_order.refl
tff(fact_3977_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_3978_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_3979_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_3980_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_3981_ent__false__iff,axiom,
    ! [P: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),bot_bot_assn))
    <=> ! [H3: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H3)) ) ).

% ent_false_iff
tff(fact_3982_snga__same__false,axiom,
    ! [P2: array_VEBT_VEBTi,X2: list_VEBT_VEBTi,Y3: list_VEBT_VEBTi] : aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(P2),X2)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(P2),Y3)) = bot_bot_assn ).

% snga_same_false
tff(fact_3983_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_3984_ceiling__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_3985_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_3986_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_3987_ceiling__le__zero,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X2)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_3988_ceiling__le__zero,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat)) ) ).

% ceiling_le_zero
tff(fact_3989_ceiling__le__numeral,axiom,
    ! [X2: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_numeral
tff(fact_3990_ceiling__le__numeral,axiom,
    ! [X2: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_numeral
tff(fact_3991_zero__less__ceiling,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2)) ) ).

% zero_less_ceiling
tff(fact_3992_zero__less__ceiling,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% zero_less_ceiling
tff(fact_3993_numeral__less__ceiling,axiom,
    ! [V: num,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,V)),X2)) ) ).

% numeral_less_ceiling
tff(fact_3994_numeral__less__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X2)) ) ).

% numeral_less_ceiling
tff(fact_3995_ceiling__less__one,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X2)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_3996_ceiling__less__one,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),zero_zero_rat)) ) ).

% ceiling_less_one
tff(fact_3997_one__le__ceiling,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2)) ) ).

% one_le_ceiling
tff(fact_3998_one__le__ceiling,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% one_le_ceiling
tff(fact_3999_ceiling__add__numeral,axiom,
    ! [X2: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4000_ceiling__add__numeral,axiom,
    ! [X2: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X2),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4001_ceiling__le__one,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X2)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ).

% ceiling_le_one
tff(fact_4002_ceiling__le__one,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),one_one_rat)) ) ).

% ceiling_le_one
tff(fact_4003_one__less__ceiling,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X2)) ) ).

% one_less_ceiling
tff(fact_4004_one__less__ceiling,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ).

% one_less_ceiling
tff(fact_4005_ceiling__add__one,axiom,
    ! [X2: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X2),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),one_one_int) ).

% ceiling_add_one
tff(fact_4006_ceiling__add__one,axiom,
    ! [X2: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X2)),one_one_int) ).

% ceiling_add_one
tff(fact_4007_ceiling__diff__numeral,axiom,
    ! [X2: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4008_ceiling__diff__numeral,axiom,
    ! [X2: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X2),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4009_ceiling__numeral__power,axiom,
    ! [X2: num,N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ).

% ceiling_numeral_power
tff(fact_4010_ceiling__numeral__power,axiom,
    ! [X2: num,N: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ).

% ceiling_numeral_power
tff(fact_4011_ceiling__diff__one,axiom,
    ! [X2: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X2),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),one_one_int) ).

% ceiling_diff_one
tff(fact_4012_ceiling__diff__one,axiom,
    ! [X2: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X2)),one_one_int) ).

% ceiling_diff_one
tff(fact_4013_ceiling__less__numeral,axiom,
    ! [X2: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_4014_ceiling__less__numeral,axiom,
    ! [X2: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_4015_numeral__le__ceiling,axiom,
    ! [V: num,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X2)) ) ).

% numeral_le_ceiling
tff(fact_4016_numeral__le__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X2)) ) ).

% numeral_le_ceiling
tff(fact_4017_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_4018_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
tff(fact_4019_ent__false,axiom,
    ! [P: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,bot_bot_assn),P)) ).

% ent_false
tff(fact_4020_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_4021_ceiling__mono,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y3)),aa_real_int(archim7802044766580827645g_real,X2))) ) ).

% ceiling_mono
tff(fact_4022_ceiling__mono,axiom,
    ! [Y3: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y3)),aa_rat_int(archim2889992004027027881ng_rat,X2))) ) ).

% ceiling_mono
tff(fact_4023_ceiling__less__cancel,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_rat_int(archim2889992004027027881ng_rat,Y3)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ).

% ceiling_less_cancel
tff(fact_4024_ceiling__less__cancel,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_real_int(archim7802044766580827645g_real,Y3)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ).

% ceiling_less_cancel
tff(fact_4025_ceiling__add__le,axiom,
    ! [X2: rat,Y3: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X2),Y3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_rat_int(archim2889992004027027881ng_rat,Y3)))) ).

% ceiling_add_le
tff(fact_4026_ceiling__add__le,axiom,
    ! [X2: real,Y3: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_real_int(archim7802044766580827645g_real,Y3)))) ).

% ceiling_add_le
tff(fact_4027_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_4028_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_4029_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_4030_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_4031_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A2)) ).

% verit_comp_simplify1(2)
tff(fact_4032_nle__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4033_nle__le,axiom,
    ! [A2: num,B2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4034_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4035_nle__le,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4036_le__cases3,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
       => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Z))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y3)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y3))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X2)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X2))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4037_le__cases3,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),Z)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Z))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y3)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y3))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),Z))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X2)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X2))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4038_le__cases3,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y3)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y3))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X2)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X2))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4039_le__cases3,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),Z)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y3)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y3))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),Z))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X2)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X2))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4040_order__class_Oorder__eq__iff,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( X2 = Y3 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_4041_order__class_Oorder__eq__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 = Y3 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_4042_order__class_Oorder__eq__iff,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 = Y3 )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_4043_order__class_Oorder__eq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 = Y3 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_4044_order__class_Oorder__eq__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 = Y3 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_4045_ord__eq__le__trans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( ( A2 = B2 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_4046_ord__eq__le__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = B2 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_4047_ord__eq__le__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( A2 = B2 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_4048_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_4049_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_4050_ord__le__eq__trans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_4051_ord__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_4052_ord__le__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_4053_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_4054_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_4055_order__antisym,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X2))
       => ( X2 = Y3 ) ) ) ).

% order_antisym
tff(fact_4056_order__antisym,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
       => ( X2 = Y3 ) ) ) ).

% order_antisym
tff(fact_4057_order__antisym,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2))
       => ( X2 = Y3 ) ) ) ).

% order_antisym
tff(fact_4058_order__antisym,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
       => ( X2 = Y3 ) ) ) ).

% order_antisym
tff(fact_4059_order__antisym,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
       => ( X2 = Y3 ) ) ) ).

% order_antisym
tff(fact_4060_order_Otrans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2)) ) ) ).

% order.trans
tff(fact_4061_order_Otrans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),C2)) ) ) ).

% order.trans
tff(fact_4062_order_Otrans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),C2)) ) ) ).

% order.trans
tff(fact_4063_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C2)) ) ) ).

% order.trans
tff(fact_4064_order_Otrans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C2)) ) ) ).

% order.trans
tff(fact_4065_order__trans,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X2),Z)) ) ) ).

% order_trans
tff(fact_4066_order__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Z)) ) ) ).

% order_trans
tff(fact_4067_order__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Z)) ) ) ).

% order_trans
tff(fact_4068_order__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Z)) ) ) ).

% order_trans
tff(fact_4069_order__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Z)) ) ) ).

% order_trans
tff(fact_4070_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A2: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
     => ( ! [A3: rat,B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B2)) ) ) ).

% linorder_wlog
tff(fact_4071_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
     => ( ! [A3: num,B3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
           => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) ) ) ).

% linorder_wlog
tff(fact_4072_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) ) ) ).

% linorder_wlog
tff(fact_4073_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) ) ) ).

% linorder_wlog
tff(fact_4074_dual__order_Oeq__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_4075_dual__order_Oeq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_4076_dual__order_Oeq__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_4077_dual__order_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_4078_dual__order_Oeq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_4079_dual__order_Oantisym,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4080_dual__order_Oantisym,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4081_dual__order_Oantisym,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4082_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4083_dual__order_Oantisym,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4084_dual__order_Otrans,axiom,
    ! [B2: set_int,A2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),B2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C2),A2)) ) ) ).

% dual_order.trans
tff(fact_4085_dual__order_Otrans,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2)) ) ) ).

% dual_order.trans
tff(fact_4086_dual__order_Otrans,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2)) ) ) ).

% dual_order.trans
tff(fact_4087_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2)) ) ) ).

% dual_order.trans
tff(fact_4088_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2)) ) ) ).

% dual_order.trans
tff(fact_4089_antisym,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4090_antisym,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4091_antisym,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4092_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4093_antisym,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4094_Orderings_Oorder__eq__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_4095_Orderings_Oorder__eq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_4096_Orderings_Oorder__eq__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_4097_Orderings_Oorder__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_4098_Orderings_Oorder__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_4099_order__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4100_order__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B2: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4101_order__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4102_order__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B2: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_int_rat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_int_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4103_order__subst1,axiom,
    ! [A2: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_rat_num(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_rat_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4104_order__subst1,axiom,
    ! [A2: num,F: fun_num_num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4105_order__subst1,axiom,
    ! [A2: num,F: fun_nat_num,B2: nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4106_order__subst1,axiom,
    ! [A2: num,F: fun_int_num,B2: int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_int_num(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_int_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4107_order__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_rat_nat(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_rat_nat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4108_order__subst1,axiom,
    ! [A2: nat,F: fun_num_nat,B2: num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_num_nat(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_num_nat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_4109_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4110_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4111_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4112_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4113_order__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4114_order__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4115_order__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4116_order__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4117_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B2)),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4118_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: fun_nat_num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B2)),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A2)),C2)) ) ) ) ).

% order_subst2
tff(fact_4119_order__eq__refl,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( X2 = Y3 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3)) ) ).

% order_eq_refl
tff(fact_4120_order__eq__refl,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 = Y3 )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% order_eq_refl
tff(fact_4121_order__eq__refl,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 = Y3 )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3)) ) ).

% order_eq_refl
tff(fact_4122_order__eq__refl,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 = Y3 )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ).

% order_eq_refl
tff(fact_4123_order__eq__refl,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 = Y3 )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ).

% order_eq_refl
tff(fact_4124_verit__la__disequality,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ).

% verit_la_disequality
tff(fact_4125_verit__la__disequality,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ).

% verit_la_disequality
tff(fact_4126_verit__la__disequality,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ).

% verit_la_disequality
tff(fact_4127_verit__la__disequality,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% verit_la_disequality
tff(fact_4128_linorder__linear,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) ) ).

% linorder_linear
tff(fact_4129_linorder__linear,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) ) ).

% linorder_linear
tff(fact_4130_linorder__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ).

% linorder_linear
tff(fact_4131_linorder__linear,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) ) ).

% linorder_linear
tff(fact_4132_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4133_ord__eq__le__subst,axiom,
    ! [A2: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_num(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_rat_num(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4134_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_nat(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_rat_nat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4135_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_rat_int,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_int(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_rat_int(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4136_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_num_rat,B2: num,C2: num] :
      ( ( A2 = aa_num_rat(F,B2) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4137_ord__eq__le__subst,axiom,
    ! [A2: num,F: fun_num_num,B2: num,C2: num] :
      ( ( A2 = aa_num_num(F,B2) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4138_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_num_nat,B2: num,C2: num] :
      ( ( A2 = aa_num_nat(F,B2) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_num_nat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4139_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_num_int,B2: num,C2: num] :
      ( ( A2 = aa_num_int(F,B2) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_num_int(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4140_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( ( A2 = aa_nat_rat(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4141_ord__eq__le__subst,axiom,
    ! [A2: num,F: fun_nat_num,B2: nat,C2: nat] :
      ( ( A2 = aa_nat_num(F,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_4142_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( ( aa_rat_rat(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4143_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( ( aa_rat_num(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4144_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( ( aa_rat_nat(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4145_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( ( aa_rat_int(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4146_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( aa_num_rat(F,B2) = C2 )
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4147_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( aa_num_num(F,B2) = C2 )
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4148_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( aa_num_nat(F,B2) = C2 )
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4149_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( aa_num_int(F,B2) = C2 )
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4150_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( aa_nat_rat(F,B2) = C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4151_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: fun_nat_num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( aa_nat_num(F,B2) = C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A2)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_4152_linorder__le__cases,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) ) ).

% linorder_le_cases
tff(fact_4153_linorder__le__cases,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) ) ).

% linorder_le_cases
tff(fact_4154_linorder__le__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ).

% linorder_le_cases
tff(fact_4155_linorder__le__cases,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) ) ).

% linorder_le_cases
tff(fact_4156_order__antisym__conv,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_4157_order__antisym__conv,axiom,
    ! [Y3: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_4158_order__antisym__conv,axiom,
    ! [Y3: num,X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_4159_order__antisym__conv,axiom,
    ! [Y3: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_4160_order__antisym__conv,axiom,
    ! [Y3: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% order_antisym_conv
tff(fact_4161_order__less__imp__not__less,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ).

% order_less_imp_not_less
tff(fact_4162_order__less__imp__not__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ).

% order_less_imp_not_less
tff(fact_4163_order__less__imp__not__less,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ).

% order_less_imp_not_less
tff(fact_4164_order__less__imp__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ).

% order_less_imp_not_less
tff(fact_4165_order__less__imp__not__less,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ).

% order_less_imp_not_less
tff(fact_4166_order__less__imp__not__eq2,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_4167_order__less__imp__not__eq2,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_4168_order__less__imp__not__eq2,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_4169_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_4170_order__less__imp__not__eq2,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_4171_order__less__imp__not__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_4172_order__less__imp__not__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_4173_order__less__imp__not__eq,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_4174_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_4175_order__less__imp__not__eq,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
tff(fact_4176_linorder__less__linear,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
      | ( X2 = Y3 )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ).

% linorder_less_linear
tff(fact_4177_linorder__less__linear,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
      | ( X2 = Y3 )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ).

% linorder_less_linear
tff(fact_4178_linorder__less__linear,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
      | ( X2 = Y3 )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ).

% linorder_less_linear
tff(fact_4179_linorder__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
      | ( X2 = Y3 )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ).

% linorder_less_linear
tff(fact_4180_linorder__less__linear,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
      | ( X2 = Y3 )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ).

% linorder_less_linear
tff(fact_4181_order__less__imp__triv,axiom,
    ! [X2: real,Y3: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_4182_order__less__imp__triv,axiom,
    ! [X2: rat,Y3: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_4183_order__less__imp__triv,axiom,
    ! [X2: num,Y3: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_4184_order__less__imp__triv,axiom,
    ! [X2: nat,Y3: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_4185_order__less__imp__triv,axiom,
    ! [X2: int,Y3: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_4186_order__less__not__sym,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ).

% order_less_not_sym
tff(fact_4187_order__less__not__sym,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ).

% order_less_not_sym
tff(fact_4188_order__less__not__sym,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ).

% order_less_not_sym
tff(fact_4189_order__less__not__sym,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ).

% order_less_not_sym
tff(fact_4190_order__less__not__sym,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ).

% order_less_not_sym
tff(fact_4191_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4192_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4193_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4194_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4195_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4196_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4197_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4198_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4199_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4200_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A2)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_4201_order__less__subst1,axiom,
    ! [A2: real,F: fun_real_real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4202_order__less__subst1,axiom,
    ! [A2: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4203_order__less__subst1,axiom,
    ! [A2: real,F: fun_num_real,B2: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4204_order__less__subst1,axiom,
    ! [A2: real,F: fun_nat_real,B2: nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_nat_real(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_nat_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4205_order__less__subst1,axiom,
    ! [A2: real,F: fun_int_real,B2: int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_int_real(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_int_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4206_order__less__subst1,axiom,
    ! [A2: rat,F: fun_real_rat,B2: real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4207_order__less__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4208_order__less__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B2: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4209_order__less__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_nat_rat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_nat_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4210_order__less__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B2: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_int_rat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_int_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_4211_order__less__irrefl,axiom,
    ! [X2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X2)) ).

% order_less_irrefl
tff(fact_4212_order__less__irrefl,axiom,
    ! [X2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),X2)) ).

% order_less_irrefl
tff(fact_4213_order__less__irrefl,axiom,
    ! [X2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),X2)) ).

% order_less_irrefl
tff(fact_4214_order__less__irrefl,axiom,
    ! [X2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),X2)) ).

% order_less_irrefl
tff(fact_4215_order__less__irrefl,axiom,
    ! [X2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),X2)) ).

% order_less_irrefl
tff(fact_4216_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ( aa_real_real(F,B2) = C2 )
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4217_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ( aa_real_rat(F,B2) = C2 )
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4218_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: fun_real_num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ( aa_real_num(F,B2) = C2 )
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4219_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: fun_real_nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ( aa_real_nat(F,B2) = C2 )
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4220_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: fun_real_int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ( aa_real_int(F,B2) = C2 )
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4221_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( ( aa_rat_real(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4222_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( ( aa_rat_rat(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4223_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( ( aa_rat_num(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4224_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( ( aa_rat_nat(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4225_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( ( aa_rat_int(F,B2) = C2 )
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A2)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_4226_ord__eq__less__subst,axiom,
    ! [A2: real,F: fun_real_real,B2: real,C2: real] :
      ( ( A2 = aa_real_real(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4227_ord__eq__less__subst,axiom,
    ! [A2: rat,F: fun_real_rat,B2: real,C2: real] :
      ( ( A2 = aa_real_rat(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4228_ord__eq__less__subst,axiom,
    ! [A2: num,F: fun_real_num,B2: real,C2: real] :
      ( ( A2 = aa_real_num(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_real_num(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4229_ord__eq__less__subst,axiom,
    ! [A2: nat,F: fun_real_nat,B2: real,C2: real] :
      ( ( A2 = aa_real_nat(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_real_nat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4230_ord__eq__less__subst,axiom,
    ! [A2: int,F: fun_real_int,B2: real,C2: real] :
      ( ( A2 = aa_real_int(F,B2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_real_int(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4231_ord__eq__less__subst,axiom,
    ! [A2: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_real(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4232_ord__eq__less__subst,axiom,
    ! [A2: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4233_ord__eq__less__subst,axiom,
    ! [A2: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_num(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_rat_num(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4234_ord__eq__less__subst,axiom,
    ! [A2: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_nat(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_rat_nat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4235_ord__eq__less__subst,axiom,
    ! [A2: int,F: fun_rat_int,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_int(F,B2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_rat_int(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_4236_order__less__trans,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z)) ) ) ).

% order_less_trans
tff(fact_4237_order__less__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z)) ) ) ).

% order_less_trans
tff(fact_4238_order__less__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z)) ) ) ).

% order_less_trans
tff(fact_4239_order__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z)) ) ) ).

% order_less_trans
tff(fact_4240_order__less__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z)) ) ) ).

% order_less_trans
tff(fact_4241_order__less__asym_H,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ).

% order_less_asym'
tff(fact_4242_order__less__asym_H,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ).

% order_less_asym'
tff(fact_4243_order__less__asym_H,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2)) ) ).

% order_less_asym'
tff(fact_4244_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2)) ) ).

% order_less_asym'
tff(fact_4245_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% order_less_asym'
tff(fact_4246_linorder__neq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ).

% linorder_neq_iff
tff(fact_4247_linorder__neq__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 != Y3 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ) ).

% linorder_neq_iff
tff(fact_4248_linorder__neq__iff,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 != Y3 )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ) ).

% linorder_neq_iff
tff(fact_4249_linorder__neq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ) ).

% linorder_neq_iff
tff(fact_4250_linorder__neq__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ) ).

% linorder_neq_iff
tff(fact_4251_order__less__asym,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ).

% order_less_asym
tff(fact_4252_order__less__asym,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ).

% order_less_asym
tff(fact_4253_order__less__asym,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ).

% order_less_asym
tff(fact_4254_order__less__asym,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ).

% order_less_asym
tff(fact_4255_order__less__asym,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ).

% order_less_asym
tff(fact_4256_linorder__neqE,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ).

% linorder_neqE
tff(fact_4257_linorder__neqE,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ) ).

% linorder_neqE
tff(fact_4258_linorder__neqE,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ) ).

% linorder_neqE
tff(fact_4259_linorder__neqE,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ) ).

% linorder_neqE
tff(fact_4260_linorder__neqE,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ) ).

% linorder_neqE
tff(fact_4261_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4262_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4263_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4264_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4265_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4266_order_Ostrict__implies__not__eq,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4267_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4268_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4269_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4270_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4271_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_4272_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_4273_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_4274_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_4275_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_4276_not__less__iff__gr__or__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4277_not__less__iff__gr__or__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2))
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4278_not__less__iff__gr__or__eq,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2))
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4279_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4280_not__less__iff__gr__or__eq,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2))
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4281_order_Ostrict__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C2)) ) ) ).

% order.strict_trans
tff(fact_4282_order_Ostrict__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C2)) ) ) ).

% order.strict_trans
tff(fact_4283_order_Ostrict__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C2)) ) ) ).

% order.strict_trans
tff(fact_4284_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C2)) ) ) ).

% order.strict_trans
tff(fact_4285_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C2)) ) ) ).

% order.strict_trans
tff(fact_4286_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A2: real,B2: real] :
      ( ! [A3: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_4287_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A2: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),A3))
       => ( ! [A3: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_4288_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_4289_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_4290_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_4291_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ? [N7: nat] :
          ( pp(aa_nat_bool(P,N7))
          & ! [M7: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N7))
             => ~ pp(aa_nat_bool(P,M7)) ) ) ) ).

% exists_least_iff
tff(fact_4292_dual__order_Oirrefl,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),A2)) ).

% dual_order.irrefl
tff(fact_4293_dual__order_Oirrefl,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),A2)) ).

% dual_order.irrefl
tff(fact_4294_dual__order_Oirrefl,axiom,
    ! [A2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),A2)) ).

% dual_order.irrefl
tff(fact_4295_dual__order_Oirrefl,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A2)) ).

% dual_order.irrefl
tff(fact_4296_dual__order_Oirrefl,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),A2)) ).

% dual_order.irrefl
tff(fact_4297_dual__order_Oasym,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% dual_order.asym
tff(fact_4298_dual__order_Oasym,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% dual_order.asym
tff(fact_4299_dual__order_Oasym,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% dual_order.asym
tff(fact_4300_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% dual_order.asym
tff(fact_4301_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% dual_order.asym
tff(fact_4302_linorder__cases,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( ( X2 != Y3 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ).

% linorder_cases
tff(fact_4303_linorder__cases,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( ( X2 != Y3 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ) ).

% linorder_cases
tff(fact_4304_linorder__cases,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( ( X2 != Y3 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ) ).

% linorder_cases
tff(fact_4305_linorder__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( ( X2 != Y3 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ) ).

% linorder_cases
tff(fact_4306_linorder__cases,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( ( X2 != Y3 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ) ).

% linorder_cases
tff(fact_4307_antisym__conv3,axiom,
    ! [Y3: real,X2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv3
tff(fact_4308_antisym__conv3,axiom,
    ! [Y3: rat,X2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv3
tff(fact_4309_antisym__conv3,axiom,
    ! [Y3: num,X2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv3
tff(fact_4310_antisym__conv3,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv3
tff(fact_4311_antisym__conv3,axiom,
    ! [Y3: int,X2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv3
tff(fact_4312_less__induct,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y5),X4))
             => pp(aa_nat_bool(P,Y5)) )
         => pp(aa_nat_bool(P,X4)) )
     => pp(aa_nat_bool(P,A2)) ) ).

% less_induct
tff(fact_4313_ord__less__eq__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_4314_ord__less__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_4315_ord__less__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_4316_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_4317_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( ( B2 = C2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_4318_ord__eq__less__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2 = B2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_4319_ord__eq__less__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = B2 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_4320_ord__eq__less__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( A2 = B2 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_4321_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_4322_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_4323_order_Oasym,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2)) ) ).

% order.asym
tff(fact_4324_order_Oasym,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2)) ) ).

% order.asym
tff(fact_4325_order_Oasym,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2)) ) ).

% order.asym
tff(fact_4326_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2)) ) ).

% order.asym
tff(fact_4327_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% order.asym
tff(fact_4328_less__imp__neq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( X2 != Y3 ) ) ).

% less_imp_neq
tff(fact_4329_less__imp__neq,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( X2 != Y3 ) ) ).

% less_imp_neq
tff(fact_4330_less__imp__neq,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( X2 != Y3 ) ) ).

% less_imp_neq
tff(fact_4331_less__imp__neq,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( X2 != Y3 ) ) ).

% less_imp_neq
tff(fact_4332_less__imp__neq,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( X2 != Y3 ) ) ).

% less_imp_neq
tff(fact_4333_dense,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y3)) ) ) ).

% dense
tff(fact_4334_dense,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ? [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y3)) ) ) ).

% dense
tff(fact_4335_gt__ex,axiom,
    ! [X2: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X_12)) ).

% gt_ex
tff(fact_4336_gt__ex,axiom,
    ! [X2: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),X_12)) ).

% gt_ex
tff(fact_4337_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),X_12)) ).

% gt_ex
tff(fact_4338_gt__ex,axiom,
    ! [X2: int] :
    ? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),X_12)) ).

% gt_ex
tff(fact_4339_lt__ex,axiom,
    ! [X2: real] :
    ? [Y4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),X2)) ).

% lt_ex
tff(fact_4340_lt__ex,axiom,
    ! [X2: rat] :
    ? [Y4: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y4),X2)) ).

% lt_ex
tff(fact_4341_lt__ex,axiom,
    ! [X2: int] :
    ? [Y4: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y4),X2)) ).

% lt_ex
tff(fact_4342_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_4343_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_4344_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_4345_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_4346_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_4347_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd3: bool,Ve3: bool,V: option4927543243414619207at_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_bool_VEBT_VEBT(vEBT_Leaf(Vd3),Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V,Va,Vb),Vc)) = bot_bot_assn ).

% vebt_assn_raw.simps(4)
tff(fact_4348_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd3: bool,Ve3: bool] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va,Vb),Vc)),vEBT_Leafi(Vd3,Ve3)) = bot_bot_assn ).

% vebt_assn_raw.simps(3)
tff(fact_4349_verit__la__generic,axiom,
    ! [A2: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X2))
      | ( A2 = X2 )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),A2)) ) ).

% verit_la_generic
tff(fact_4350_mult__ceiling__le,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le
tff(fact_4351_mult__ceiling__le,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A2),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A2)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le
tff(fact_4352_order__le__imp__less__or__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4353_order__le__imp__less__or__eq,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
     => ( pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4354_order__le__imp__less__or__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4355_order__le__imp__less__or__eq,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4356_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4357_order__le__imp__less__or__eq,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4358_linorder__le__less__linear,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ).

% linorder_le_less_linear
tff(fact_4359_linorder__le__less__linear,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ).

% linorder_le_less_linear
tff(fact_4360_linorder__le__less__linear,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ).

% linorder_le_less_linear
tff(fact_4361_linorder__le__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ).

% linorder_le_less_linear
tff(fact_4362_linorder__le__less__linear,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ).

% linorder_le_less_linear
tff(fact_4363_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4364_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4365_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_real,C2: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4366_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: fun_nat_real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B2)),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4367_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F: fun_int_real,C2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B2)),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4368_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B2)),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4369_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4370_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4371_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B2)),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4372_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F: fun_int_rat,C2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B2)),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A2)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_4373_order__less__le__subst1,axiom,
    ! [A2: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4374_order__less__le__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4375_order__less__le__subst1,axiom,
    ! [A2: num,F: fun_rat_num,B2: rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_rat_num(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_rat_num(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4376_order__less__le__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B2: rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_rat_nat(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_rat_nat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4377_order__less__le__subst1,axiom,
    ! [A2: int,F: fun_rat_int,B2: rat,C2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_rat_int(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_rat_int(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4378_order__less__le__subst1,axiom,
    ! [A2: real,F: fun_num_real,B2: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4379_order__less__le__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B2: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4380_order__less__le__subst1,axiom,
    ! [A2: num,F: fun_num_num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_num_num(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_num_num(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4381_order__less__le__subst1,axiom,
    ! [A2: nat,F: fun_num_nat,B2: num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_num_nat(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_num_nat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4382_order__less__le__subst1,axiom,
    ! [A2: int,F: fun_num_int,B2: num,C2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_num_int(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_num_int(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_4383_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4384_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4385_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4386_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4387_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B2)),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4388_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_real,C2: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4389_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4390_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4391_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4392_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B2)),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A2)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_4393_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_real_real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_real_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4394_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_rat_real,B2: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_rat_real(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_rat_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4395_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_num_real,B2: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_num_real(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_num_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4396_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_nat_real,B2: nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_nat_real(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_nat_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4397_order__le__less__subst1,axiom,
    ! [A2: real,F: fun_int_real,B2: int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_int_real(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),aa_int_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4398_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_real_rat,B2: real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_real_rat(F,B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => ( ! [X4: real,Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_real_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4399_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(F,B2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => ( ! [X4: rat,Y4: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_rat_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4400_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_num_rat,B2: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_num_rat(F,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => ( ! [X4: num,Y4: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_num_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4401_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B2: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_nat_rat(F,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_nat_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4402_order__le__less__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B2: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_int_rat(F,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => ( ! [X4: int,Y4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),aa_int_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_4403_order__less__le__trans,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_4404_order__less__le__trans,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_4405_order__less__le__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_4406_order__less__le__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_4407_order__less__le__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_4408_order__less__le__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_4409_order__le__less__trans,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_4410_order__le__less__trans,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y3),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_4411_order__le__less__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_4412_order__le__less__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_4413_order__le__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_4414_order__le__less__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_4415_order__neq__le__trans,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != B2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_4416_order__neq__le__trans,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 != B2 )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_set_int_bool(ord_less_set_int(A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_4417_order__neq__le__trans,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != B2 )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_4418_order__neq__le__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 != B2 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_4419_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_4420_order__neq__le__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_4421_order__le__neq__trans,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_4422_order__le__neq__trans,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_set_int_bool(ord_less_set_int(A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_4423_order__le__neq__trans,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_4424_order__le__neq__trans,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_4425_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_4426_order__le__neq__trans,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_4427_order__less__imp__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% order_less_imp_le
tff(fact_4428_order__less__imp__le,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3)) ) ).

% order_less_imp_le
tff(fact_4429_order__less__imp__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% order_less_imp_le
tff(fact_4430_order__less__imp__le,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3)) ) ).

% order_less_imp_le
tff(fact_4431_order__less__imp__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ).

% order_less_imp_le
tff(fact_4432_order__less__imp__le,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ).

% order_less_imp_le
tff(fact_4433_linorder__not__less,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2)) ) ).

% linorder_not_less
tff(fact_4434_linorder__not__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) ) ).

% linorder_not_less
tff(fact_4435_linorder__not__less,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) ) ).

% linorder_not_less
tff(fact_4436_linorder__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ).

% linorder_not_less
tff(fact_4437_linorder__not__less,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) ) ).

% linorder_not_less
tff(fact_4438_linorder__not__le,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ).

% linorder_not_le
tff(fact_4439_linorder__not__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ) ).

% linorder_not_le
tff(fact_4440_linorder__not__le,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y3),X2)) ) ).

% linorder_not_le
tff(fact_4441_linorder__not__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2)) ) ).

% linorder_not_le
tff(fact_4442_linorder__not__le,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X2)) ) ).

% linorder_not_le
tff(fact_4443_order__less__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
        & ( X2 != Y3 ) ) ) ).

% order_less_le
tff(fact_4444_order__less__le,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
        & ( X2 != Y3 ) ) ) ).

% order_less_le
tff(fact_4445_order__less__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
        & ( X2 != Y3 ) ) ) ).

% order_less_le
tff(fact_4446_order__less__le,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
        & ( X2 != Y3 ) ) ) ).

% order_less_le
tff(fact_4447_order__less__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
        & ( X2 != Y3 ) ) ) ).

% order_less_le
tff(fact_4448_order__less__le,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
        & ( X2 != Y3 ) ) ) ).

% order_less_le
tff(fact_4449_order__le__less,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_less
tff(fact_4450_order__le__less,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_less
tff(fact_4451_order__le__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_less
tff(fact_4452_order__le__less,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_less
tff(fact_4453_order__le__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_less
tff(fact_4454_order__le__less,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% order_le_less
tff(fact_4455_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_4456_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_4457_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_4458_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_4459_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_4460_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_4461_order_Ostrict__implies__order,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_4462_order_Ostrict__implies__order,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% order.strict_implies_order
tff(fact_4463_order_Ostrict__implies__order,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_4464_order_Ostrict__implies__order,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_4465_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_4466_order_Ostrict__implies__order,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_4467_dual__order_Ostrict__iff__not,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_4468_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_4469_dual__order_Ostrict__iff__not,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_4470_dual__order_Ostrict__iff__not,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_4471_dual__order_Ostrict__iff__not,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_4472_dual__order_Ostrict__iff__not,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_4473_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_4474_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_int,A2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),B2))
       => pp(aa_set_int_bool(ord_less_set_int(C2),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_4475_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_4476_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_4477_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_4478_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_4479_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_4480_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_int,A2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(ord_less_set_int(C2),B2))
       => pp(aa_set_int_bool(ord_less_set_int(C2),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_4481_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_4482_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_4483_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_4484_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_4485_dual__order_Ostrict__iff__order,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4486_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B2),A2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4487_dual__order_Ostrict__iff__order,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4488_dual__order_Ostrict__iff__order,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4489_dual__order_Ostrict__iff__order,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4490_dual__order_Ostrict__iff__order,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4491_dual__order_Oorder__iff__strict,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4492_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4493_dual__order_Oorder__iff__strict,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4494_dual__order_Oorder__iff__strict,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4495_dual__order_Oorder__iff__strict,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4496_dual__order_Oorder__iff__strict,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4497_dense__le__bounded,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( ! [W2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),W2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z)) ) ) ).

% dense_le_bounded
tff(fact_4498_dense__le__bounded,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( ! [W2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),W2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z)) ) ) ).

% dense_le_bounded
tff(fact_4499_dense__ge__bounded,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( ! [W2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),X2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),W2)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z)) ) ) ).

% dense_ge_bounded
tff(fact_4500_dense__ge__bounded,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ( ! [W2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),X2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),W2)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z)) ) ) ).

% dense_ge_bounded
tff(fact_4501_order_Ostrict__iff__not,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_4502_order_Ostrict__iff__not,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_4503_order_Ostrict__iff__not,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_4504_order_Ostrict__iff__not,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_4505_order_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_4506_order_Ostrict__iff__not,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_4507_order_Ostrict__trans2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C2)) ) ) ).

% order.strict_trans2
tff(fact_4508_order_Ostrict__trans2,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C2)) ) ) ).

% order.strict_trans2
tff(fact_4509_order_Ostrict__trans2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C2)) ) ) ).

% order.strict_trans2
tff(fact_4510_order_Ostrict__trans2,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C2)) ) ) ).

% order.strict_trans2
tff(fact_4511_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C2)) ) ) ).

% order.strict_trans2
tff(fact_4512_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C2)) ) ) ).

% order.strict_trans2
tff(fact_4513_order_Ostrict__trans1,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C2)) ) ) ).

% order.strict_trans1
tff(fact_4514_order_Ostrict__trans1,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_set_int(B2),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C2)) ) ) ).

% order.strict_trans1
tff(fact_4515_order_Ostrict__trans1,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),C2)) ) ) ).

% order.strict_trans1
tff(fact_4516_order_Ostrict__trans1,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C2)) ) ) ).

% order.strict_trans1
tff(fact_4517_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C2)) ) ) ).

% order.strict_trans1
tff(fact_4518_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C2)) ) ) ).

% order.strict_trans1
tff(fact_4519_order_Ostrict__iff__order,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4520_order_Ostrict__iff__order,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4521_order_Ostrict__iff__order,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4522_order_Ostrict__iff__order,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4523_order_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4524_order_Ostrict__iff__order,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4525_order_Oorder__iff__strict,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4526_order_Oorder__iff__strict,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4527_order_Oorder__iff__strict,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4528_order_Oorder__iff__strict,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4529_order_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4530_order_Oorder__iff__strict,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4531_not__le__imp__less,axiom,
    ! [Y3: real,X2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ).

% not_le_imp_less
tff(fact_4532_not__le__imp__less,axiom,
    ! [Y3: rat,X2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ).

% not_le_imp_less
tff(fact_4533_not__le__imp__less,axiom,
    ! [Y3: num,X2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3)) ) ).

% not_le_imp_less
tff(fact_4534_not__le__imp__less,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ).

% not_le_imp_less
tff(fact_4535_not__le__imp__less,axiom,
    ! [Y3: int,X2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3)) ) ).

% not_le_imp_less
tff(fact_4536_less__le__not__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2)) ) ) ).

% less_le_not_le
tff(fact_4537_less__le__not__le,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X2)) ) ) ).

% less_le_not_le
tff(fact_4538_less__le__not__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) ) ) ).

% less_le_not_le
tff(fact_4539_less__le__not__le,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) ) ) ).

% less_le_not_le
tff(fact_4540_less__le__not__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ) ).

% less_le_not_le
tff(fact_4541_less__le__not__le,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) ) ) ).

% less_le_not_le
tff(fact_4542_dense__le,axiom,
    ! [Y3: real,Z: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z)) ) ).

% dense_le
tff(fact_4543_dense__le,axiom,
    ! [Y3: rat,Z: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Z)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z)) ) ).

% dense_le
tff(fact_4544_dense__ge,axiom,
    ! [Z: real,Y3: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z)) ) ).

% dense_ge
tff(fact_4545_dense__ge,axiom,
    ! [Z: rat,Y3: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),Z)) ) ).

% dense_ge
tff(fact_4546_antisym__conv2,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv2
tff(fact_4547_antisym__conv2,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv2
tff(fact_4548_antisym__conv2,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv2
tff(fact_4549_antisym__conv2,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv2
tff(fact_4550_antisym__conv2,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv2
tff(fact_4551_antisym__conv2,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv2
tff(fact_4552_antisym__conv1,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv1
tff(fact_4553_antisym__conv1,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X2),Y3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv1
tff(fact_4554_antisym__conv1,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv1
tff(fact_4555_antisym__conv1,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv1
tff(fact_4556_antisym__conv1,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv1
tff(fact_4557_antisym__conv1,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
      <=> ( X2 = Y3 ) ) ) ).

% antisym_conv1
tff(fact_4558_nless__le,axiom,
    ! [A2: real,B2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4559_nless__le,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4560_nless__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4561_nless__le,axiom,
    ! [A2: num,B2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4562_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4563_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4564_leI,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2)) ) ).

% leI
tff(fact_4565_leI,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) ) ).

% leI
tff(fact_4566_leI,axiom,
    ! [X2: num,Y3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) ) ).

% leI
tff(fact_4567_leI,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ).

% leI
tff(fact_4568_leI,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) ) ).

% leI
tff(fact_4569_leD,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ).

% leD
tff(fact_4570_leD,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X2))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X2),Y3)) ) ).

% leD
tff(fact_4571_leD,axiom,
    ! [Y3: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ).

% leD
tff(fact_4572_leD,axiom,
    ! [Y3: num,X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y3)) ) ).

% leD
tff(fact_4573_leD,axiom,
    ! [Y3: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3)) ) ).

% leD
tff(fact_4574_leD,axiom,
    ! [Y3: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3)) ) ).

% leD
tff(fact_4575_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),A4))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_4576_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B4),A4))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_4577_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B4),A4))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_4578_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B4),A4))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_4579_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),A4))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_4580_verit__sum__simplify,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),zero_zero_complex) = A2 ).

% verit_sum_simplify
tff(fact_4581_verit__sum__simplify,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% verit_sum_simplify
tff(fact_4582_verit__sum__simplify,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),zero_zero_rat) = A2 ).

% verit_sum_simplify
tff(fact_4583_verit__sum__simplify,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% verit_sum_simplify
tff(fact_4584_verit__sum__simplify,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% verit_sum_simplify
tff(fact_4585_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_4586_bot_Oextremum,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(bot_bot_set_nat),A2)) ).

% bot.extremum
tff(fact_4587_bot_Oextremum,axiom,
    ! [A2: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(ord_less_eq_assn,bot_bot_assn),A2)) ).

% bot.extremum
tff(fact_4588_bot_Oextremum,axiom,
    ! [A2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(bot_bo4199563552545308370d_enat),A2)) ).

% bot.extremum
tff(fact_4589_bot_Oextremum,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A2)) ).

% bot.extremum
tff(fact_4590_bot_Oextremum,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A2)) ).

% bot.extremum
tff(fact_4591_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),bot_bot_set_nat))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_4592_bot_Oextremum__unique,axiom,
    ! [A2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(ord_less_eq_assn,A2),bot_bot_assn))
    <=> ( A2 = bot_bot_assn ) ) ).

% bot.extremum_unique
tff(fact_4593_bot_Oextremum__unique,axiom,
    ! [A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),bot_bo4199563552545308370d_enat))
    <=> ( A2 = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_unique
tff(fact_4594_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
    <=> ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_4595_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),bot_bot_nat))
    <=> ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_4596_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),bot_bot_set_nat))
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_4597_bot_Oextremum__uniqueI,axiom,
    ! [A2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(ord_less_eq_assn,A2),bot_bot_assn))
     => ( A2 = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
tff(fact_4598_bot_Oextremum__uniqueI,axiom,
    ! [A2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),bot_bo4199563552545308370d_enat))
     => ( A2 = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_uniqueI
tff(fact_4599_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_4600_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),bot_bot_nat))
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_4601_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_set_nat(bot_bot_set_nat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_4602_bot_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != bot_bot_assn )
    <=> pp(aa_assn_bool(aa_ass5323576183532441186n_bool(ord_less_assn,bot_bot_assn),A2)) ) ).

% bot.not_eq_extremum
tff(fact_4603_bot_Onot__eq__extremum,axiom,
    ! [A2: extended_enat] :
      ( ( A2 != bot_bo4199563552545308370d_enat )
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(bot_bo4199563552545308370d_enat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_4604_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A2)) ) ).

% bot.not_eq_extremum
tff(fact_4605_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_4606_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A2),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_4607_bot_Oextremum__strict,axiom,
    ! [A2: assn] : ~ pp(aa_assn_bool(aa_ass5323576183532441186n_bool(ord_less_assn,A2),bot_bot_assn)) ).

% bot.extremum_strict
tff(fact_4608_bot_Oextremum__strict,axiom,
    ! [A2: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A2),bot_bo4199563552545308370d_enat)) ).

% bot.extremum_strict
tff(fact_4609_bot_Oextremum__strict,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_4610_bot_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_4611_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_4612_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one != aa_num_num(bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_4613_max__def,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),B2))
       => ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A2),B2))
       => ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ) ).

% max_def
tff(fact_4614_max__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ord_max_set_int(A2,B2) = B2 ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ord_max_set_int(A2,B2) = A2 ) ) ) ).

% max_def
tff(fact_4615_max__def,axiom,
    ! [A2: rat,B2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ) ).

% max_def
tff(fact_4616_max__def,axiom,
    ! [A2: num,B2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
       => ( aa_num_num(ord_max_num(A2),B2) = B2 ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
       => ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ) ).

% max_def
tff(fact_4617_max__def,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ) ).

% max_def
tff(fact_4618_max__def,axiom,
    ! [A2: int,B2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ) ).

% max_def
tff(fact_4619_max__absorb1,axiom,
    ! [Y3: extended_enat,X2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Y3),X2))
     => ( ord_ma741700101516333627d_enat(X2,Y3) = X2 ) ) ).

% max_absorb1
tff(fact_4620_max__absorb1,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X2))
     => ( ord_max_set_int(X2,Y3) = X2 ) ) ).

% max_absorb1
tff(fact_4621_max__absorb1,axiom,
    ! [Y3: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2))
     => ( aa_rat_rat(ord_max_rat(X2),Y3) = X2 ) ) ).

% max_absorb1
tff(fact_4622_max__absorb1,axiom,
    ! [Y3: num,X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2))
     => ( aa_num_num(ord_max_num(X2),Y3) = X2 ) ) ).

% max_absorb1
tff(fact_4623_max__absorb1,axiom,
    ! [Y3: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3) = X2 ) ) ).

% max_absorb1
tff(fact_4624_max__absorb1,axiom,
    ! [Y3: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3) = X2 ) ) ).

% max_absorb1
tff(fact_4625_max__absorb2,axiom,
    ! [X2: extended_enat,Y3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X2),Y3))
     => ( ord_ma741700101516333627d_enat(X2,Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_4626_max__absorb2,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3))
     => ( ord_max_set_int(X2,Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_4627_max__absorb2,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => ( aa_rat_rat(ord_max_rat(X2),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_4628_max__absorb2,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3))
     => ( aa_num_num(ord_max_num(X2),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_4629_max__absorb2,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_4630_max__absorb2,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3) = Y3 ) ) ).

% max_absorb2
tff(fact_4631_TBOUND__mono,axiom,
    ! [C2: heap_T8145700208782473153_VEBTi,T: nat,T3: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(C2),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T3))
       => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(C2),T3)) ) ) ).

% TBOUND_mono
tff(fact_4632_TBOUND__mono,axiom,
    ! [C2: heap_Time_Heap_bool,T: nat,T3: nat] :
      ( pp(aa_nat_bool(time_TBOUND_bool(C2),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T3))
       => pp(aa_nat_bool(time_TBOUND_bool(C2),T3)) ) ) ).

% TBOUND_mono
tff(fact_4633_TBOUND__mono,axiom,
    ! [C2: heap_T2636463487746394924on_nat,T: nat,T3: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(C2),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T3))
       => pp(aa_nat_bool(time_T8353473612707095248on_nat(C2),T3)) ) ) ).

% TBOUND_mono
tff(fact_4634_TBOUND__mono,axiom,
    ! [C2: heap_Time_Heap_nat,T: nat,T3: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(C2),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T3))
       => pp(aa_nat_bool(time_TBOUND_nat(C2),T3)) ) ) ).

% TBOUND_mono
tff(fact_4635_TBOUND__if__max,axiom,
    ! [P: bool,M: heap_T8145700208782473153_VEBTi,Bm: nat,N: heap_T8145700208782473153_VEBTi,Bn: nat] :
      ( ( pp(P)
       => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),Bm)) )
     => ( ( ~ pp(P)
         => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(N),Bn)) )
       => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(if_Hea8453224502484754311_VEBTi(P,M,N)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Bm),Bn))) ) ) ).

% TBOUND_if_max
tff(fact_4636_TBOUND__if__max,axiom,
    ! [P: bool,M: heap_Time_Heap_bool,Bm: nat,N: heap_Time_Heap_bool,Bn: nat] :
      ( ( pp(P)
       => pp(aa_nat_bool(time_TBOUND_bool(M),Bm)) )
     => ( ( ~ pp(P)
         => pp(aa_nat_bool(time_TBOUND_bool(N),Bn)) )
       => pp(aa_nat_bool(time_TBOUND_bool(if_Hea7696252282596813915p_bool(P,M,N)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Bm),Bn))) ) ) ).

% TBOUND_if_max
tff(fact_4637_TBOUND__if__max,axiom,
    ! [P: bool,M: heap_T2636463487746394924on_nat,Bm: nat,N: heap_T2636463487746394924on_nat,Bn: nat] :
      ( ( pp(P)
       => pp(aa_nat_bool(time_T8353473612707095248on_nat(M),Bm)) )
     => ( ( ~ pp(P)
         => pp(aa_nat_bool(time_T8353473612707095248on_nat(N),Bn)) )
       => pp(aa_nat_bool(time_T8353473612707095248on_nat(if_Hea5867803462524415986on_nat(P,M,N)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Bm),Bn))) ) ) ).

% TBOUND_if_max
tff(fact_4638_TBOUND__if__max,axiom,
    ! [P: bool,M: heap_Time_Heap_nat,Bm: nat,N: heap_Time_Heap_nat,Bn: nat] :
      ( ( pp(P)
       => pp(aa_nat_bool(time_TBOUND_nat(M),Bm)) )
     => ( ( ~ pp(P)
         => pp(aa_nat_bool(time_TBOUND_nat(N),Bn)) )
       => pp(aa_nat_bool(time_TBOUND_nat(if_Hea2662716070787841314ap_nat(P,M,N)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Bm),Bn))) ) ) ).

% TBOUND_if_max
tff(fact_4639_max__def__raw,axiom,
    ! [X6: extended_enat,Xa2: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X6),Xa2))
       => ( ord_ma741700101516333627d_enat(X6,Xa2) = Xa2 ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X6),Xa2))
       => ( ord_ma741700101516333627d_enat(X6,Xa2) = X6 ) ) ) ).

% max_def_raw
tff(fact_4640_max__def__raw,axiom,
    ! [X6: set_int,Xa2: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),Xa2))
       => ( ord_max_set_int(X6,Xa2) = Xa2 ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X6),Xa2))
       => ( ord_max_set_int(X6,Xa2) = X6 ) ) ) ).

% max_def_raw
tff(fact_4641_max__def__raw,axiom,
    ! [X6: rat,Xa2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X6),Xa2))
       => ( aa_rat_rat(ord_max_rat(X6),Xa2) = Xa2 ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X6),Xa2))
       => ( aa_rat_rat(ord_max_rat(X6),Xa2) = X6 ) ) ) ).

% max_def_raw
tff(fact_4642_max__def__raw,axiom,
    ! [X6: num,Xa2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X6),Xa2))
       => ( aa_num_num(ord_max_num(X6),Xa2) = Xa2 ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X6),Xa2))
       => ( aa_num_num(ord_max_num(X6),Xa2) = X6 ) ) ) ).

% max_def_raw
tff(fact_4643_max__def__raw,axiom,
    ! [X6: nat,Xa2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),Xa2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X6),Xa2) = Xa2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),Xa2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X6),Xa2) = X6 ) ) ) ).

% max_def_raw
tff(fact_4644_max__def__raw,axiom,
    ! [X6: int,Xa2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),Xa2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X6),Xa2) = Xa2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),Xa2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X6),Xa2) = X6 ) ) ) ).

% max_def_raw
tff(fact_4645_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_4646_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) ) ).

% nat_int_comparison(2)
tff(fact_4647_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) ) ).

% nat_int_comparison(3)
tff(fact_4648_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_4649_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_4650_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(5)
tff(fact_4651_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(7)
tff(fact_4652_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H)),T)) ) ).

% TBOUNDD
tff(fact_4653_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_bool,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_TBOUND_bool(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(M,H)),T)) ) ).

% TBOUNDD
tff(fact_4654_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(M,H)),T)) ) ).

% TBOUNDD
tff(fact_4655_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_TBOUND_nat(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(M,H)),T)) ) ).

% TBOUNDD
tff(fact_4656_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H2)),T))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),T)) ) ).

% TBOUNDI
tff(fact_4657_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_bool,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(M,H2)),T))
     => pp(aa_nat_bool(time_TBOUND_bool(M),T)) ) ).

% TBOUNDI
tff(fact_4658_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(M,H2)),T))
     => pp(aa_nat_bool(time_T8353473612707095248on_nat(M),T)) ) ).

% TBOUNDI
tff(fact_4659_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(M,H2)),T))
     => pp(aa_nat_bool(time_TBOUND_nat(M),T)) ) ).

% TBOUNDI
tff(fact_4660_TBOUND__def,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_4661_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_bool,T: nat] :
      ( pp(aa_nat_bool(time_TBOUND_bool(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_4662_TBOUND__def,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_4663_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_4664_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_4665_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_4666_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B2)),P),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4667_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: bool,P: assn,F: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B2)),P),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_bool(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4668_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B2)),P),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_option_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4669_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B2)),P),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4670_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4671_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_bool(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4672_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_option_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4673_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B2)),F,Q),T))
    <=> ( pp(B2)
       => pp(aa_nat_bool(time_htt_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4674_nat__less__as__int,axiom,
    ! [X6: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ) ).

% nat_less_as_int
tff(fact_4675_nat__leq__as__int,axiom,
    ! [X6: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ) ).

% nat_leq_as_int
tff(fact_4676_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_4677_htt__cons__rule,axiom,
    ! [P3: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,T3: nat,P: assn,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(P3,C2,Q2),T3))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
       => ( ! [X4: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,X4)),aa_VEBT_VEBTi_assn(Q,X4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T3),T))
           => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C2,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4678_htt__cons__rule,axiom,
    ! [P3: assn,C2: heap_Time_Heap_bool,Q2: fun_bool_assn,T3: nat,P: assn,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(P3,C2,Q2),T3))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
       => ( ! [X4: bool] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_bool_assn(Q2,X4)),aa_bool_assn(Q,X4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T3),T))
           => pp(aa_nat_bool(time_htt_bool(P,C2,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4679_htt__cons__rule,axiom,
    ! [P3: assn,C2: heap_T2636463487746394924on_nat,Q2: fun_option_nat_assn,T3: nat,P: assn,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(P3,C2,Q2),T3))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
       => ( ! [X4: option_nat] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_option_nat_assn(Q2,X4)),aa_option_nat_assn(Q,X4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T3),T))
           => pp(aa_nat_bool(time_htt_option_nat(P,C2,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4680_htt__cons__rule,axiom,
    ! [P3: assn,C2: heap_Time_Heap_nat,Q2: fun_nat_assn,T3: nat,P: assn,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(P3,C2,Q2),T3))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P3))
       => ( ! [X4: nat] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_nat_assn(Q2,X4)),aa_nat_assn(Q,X4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T3),T))
           => pp(aa_nat_bool(time_htt_nat(P,C2,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4681_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_4682_int__ops_I4_J,axiom,
    ! [A2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),one_one_int) ).

% int_ops(4)
tff(fact_4683_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M)) ) ).

% less_1_helper
tff(fact_4684_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(time_htt_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_memberi(Ti,X2),aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_bm(T),Ti),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_4685_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_inserti(Ti,X2),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_4686_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_predi(Ti,X2),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_bn(T),Ti),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))))))) ) ).

% htt_vebt_predi
tff(fact_4687_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_succi(Ti,X2),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_bo(T),Ti),X2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))))))) ) ).

% htt_vebt_succi
tff(fact_4688_log__ceil__idem,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X2)))) ) ) ).

% log_ceil_idem
tff(fact_4689_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                 => ! [N3: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y3 = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( Y3 = one_one_nat ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y3 = one_one_nat ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_4690_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_4691_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_4692_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_4693_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_4694_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4695_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4696_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4697_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4698_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_4699_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_4700_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_4701_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_4702_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_4703_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_4704_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4705_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4706_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4707_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_4708_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4709_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_4710_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_4711_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_4712_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_4713_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_4714_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_4715_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_4716_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_4717_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_4718_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_4719_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_4720_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X2) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) )
    <=> ( X2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4721_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) )
    <=> ( X2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4722_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B2)),W) )
    <=> ( X2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4723_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B2)),W) )
    <=> ( X2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4724_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,X2) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W) )
    <=> ( X2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4725_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) = aa_int_rat(ring_1_of_int_rat,X2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4726_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) = aa_int_real(ring_1_of_int_real,X2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4727_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B2)),W) = aa_int_int(ring_1_of_int_int,X2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4728_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B2)),W) = aa_int_complex(ring_17405671764205052669omplex,X2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4729_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W) = aa_int_Code_integer(ring_18347121197199848620nteger,X2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4730_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_4731_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_4732_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_4733_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).

% of_int_power
tff(fact_4734_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)),N) ).

% of_int_power
tff(fact_4735_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_4736_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_4737_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_4738_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% zless_nat_conj
tff(fact_4739_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_4740_ceiling__add__of__int,axiom,
    ! [X2: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X2),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),Z) ).

% ceiling_add_of_int
tff(fact_4741_ceiling__add__of__int,axiom,
    ! [X2: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X2)),Z) ).

% ceiling_add_of_int
tff(fact_4742_ceiling__diff__of__int,axiom,
    ! [X2: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X2),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),Z) ).

% ceiling_diff_of_int
tff(fact_4743_ceiling__diff__of__int,axiom,
    ! [X2: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X2)),Z) ).

% ceiling_diff_of_int
tff(fact_4744_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_4745_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_4746_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_4747_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_4748_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_4749_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_4750_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_4751_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_4752_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_4753_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_4754_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_4755_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_4756_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_4757_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_4758_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_4759_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_4760_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_4761_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_4762_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_4763_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_4764_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_4765_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_4766_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_4767_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_4768_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_4769_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_4770_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_4771_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_4772_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_4773_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_4774_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_4775_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_4776_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_4777_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_4778_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_4779_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_4780_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X2: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y3) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4781_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X2: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y3) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X2)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4782_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X2: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4783_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X2: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y3) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4784_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X2: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4785_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4786_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X2)),N) = aa_int_complex(ring_17405671764205052669omplex,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4787_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N) = aa_int_real(ring_1_of_int_real,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4788_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N) = aa_int_rat(ring_1_of_int_rat,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4789_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = aa_int_int(ring_1_of_int_int,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4790_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4791_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4792_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4793_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4794_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4795_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4796_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4797_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4798_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4799_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4800_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X2)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4801_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B2: int,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B2)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4802_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4803_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4804_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4805_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),W)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4806_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_4807_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_4808_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).

% of_nat_nat
tff(fact_4809_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_4810_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_4811_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y3: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = aa_int_nat(nat2,Y3) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_4812_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X2: num,N: nat] :
      ( ( aa_int_nat(nat2,Y3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) )
    <=> ( Y3 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_4813_nat__ceiling__le__eq,axiom,
    ! [X2: real,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X2))),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(semiri5074537144036343181t_real,A2))) ) ).

% nat_ceiling_le_eq
tff(fact_4814_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_4815_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4816_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N)),aa_int_real(ring_1_of_int_real,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4817_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N)),aa_int_rat(ring_1_of_int_rat,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4818_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),aa_int_int(ring_1_of_int_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4819_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4820_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4821_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4822_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4823_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4824_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4825_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4826_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4827_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4828_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N)),aa_int_real(ring_1_of_int_real,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4829_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N)),aa_int_rat(ring_1_of_int_rat,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4830_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),aa_int_int(ring_1_of_int_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4831_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_4832_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N)),aa_int_nat(nat2,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4833_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4834_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4835_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N)),aa_int_nat(nat2,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N)),A2)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4836_ex__le__of__int,axiom,
    ! [X2: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_le_of_int
tff(fact_4837_ex__le__of__int,axiom,
    ! [X2: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_le_of_int
tff(fact_4838_ex__of__int__less,axiom,
    ! [X2: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X2)) ).

% ex_of_int_less
tff(fact_4839_ex__of__int__less,axiom,
    ! [X2: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X2)) ).

% ex_of_int_less
tff(fact_4840_ex__less__of__int,axiom,
    ! [X2: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_less_of_int
tff(fact_4841_ex__less__of__int,axiom,
    ! [X2: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_less_of_int
tff(fact_4842_mult__of__int__commute,axiom,
    ! [X2: int,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X2)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),aa_int_real(ring_1_of_int_real,X2)) ).

% mult_of_int_commute
tff(fact_4843_mult__of__int__commute,axiom,
    ! [X2: int,Y3: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X2)),Y3) = aa_rat_rat(times_times_rat(Y3),aa_int_rat(ring_1_of_int_rat,X2)) ).

% mult_of_int_commute
tff(fact_4844_mult__of__int__commute,axiom,
    ! [X2: int,Y3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X2)),Y3) = aa_int_int(aa_int_fun_int_int(times_times_int,Y3),aa_int_int(ring_1_of_int_int,X2)) ).

% mult_of_int_commute
tff(fact_4845_of__int__max,axiom,
    ! [X2: int,Y3: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_int_real(ring_1_of_int_real,X2)),aa_int_real(ring_1_of_int_real,Y3)) ).

% of_int_max
tff(fact_4846_of__int__max,axiom,
    ! [X2: int,Y3: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,X2)),aa_int_rat(ring_1_of_int_rat,Y3)) ).

% of_int_max
tff(fact_4847_of__int__max,axiom,
    ! [X2: int,Y3: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,X2)),aa_int_int(ring_1_of_int_int,Y3)) ).

% of_int_max
tff(fact_4848_nat__numeral__as__int,axiom,
    ! [X6: num] : aa_num_nat(numeral_numeral_nat,X6) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X6)) ).

% nat_numeral_as_int
tff(fact_4849_nat__mono,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y3))) ) ).

% nat_mono
tff(fact_4850_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_4851_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% ex_nat
tff(fact_4852_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% all_nat
tff(fact_4853_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_4854_le__of__int__ceiling,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X2)))) ).

% le_of_int_ceiling
tff(fact_4855_le__of__int__ceiling,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X2)))) ).

% le_of_int_ceiling
tff(fact_4856_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_4857_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_4858_nat__le__iff,axiom,
    ! [X2: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X2)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_4859_of__nat__ceiling,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2))))) ).

% of_nat_ceiling
tff(fact_4860_of__nat__ceiling,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R2))))) ).

% of_nat_ceiling
tff(fact_4861_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) ).

% nat_int_add
tff(fact_4862_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_4863_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_4864_ceiling__le__iff,axiom,
    ! [X2: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X2)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,Z))) ) ).

% ceiling_le_iff
tff(fact_4865_ceiling__le__iff,axiom,
    ! [X2: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% ceiling_le_iff
tff(fact_4866_ceiling__le,axiom,
    ! [X2: real,A2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,A2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X2)),A2)) ) ).

% ceiling_le
tff(fact_4867_ceiling__le,axiom,
    ! [X2: rat,A2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,A2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),A2)) ) ).

% ceiling_le
tff(fact_4868_less__ceiling__iff,axiom,
    ! [Z: int,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X2)) ) ).

% less_ceiling_iff
tff(fact_4869_less__ceiling__iff,axiom,
    ! [Z: int,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X2)) ) ).

% less_ceiling_iff
tff(fact_4870_int__minus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_4871_real__of__int__div4,axiom,
    ! [N: int,X2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X2)))) ).

% real_of_int_div4
tff(fact_4872_real__nat__ceiling__ge,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X2))))) ).

% real_nat_ceiling_ge
tff(fact_4873_nat__plus__as__int,axiom,
    ! [X6: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_4874_nat__times__as__int,axiom,
    ! [X6: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X6),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_times_as_int
tff(fact_4875_nat__minus__as__int,axiom,
    ! [X6: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X6),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_minus_as_int
tff(fact_4876_nat__div__as__int,axiom,
    ! [X6: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X6),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_4877_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_nonneg
tff(fact_4878_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_nonneg
tff(fact_4879_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_nonneg
tff(fact_4880_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_pos
tff(fact_4881_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_pos
tff(fact_4882_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_pos
tff(fact_4883_floor__exists,axiom,
    ! [X2: real] :
    ? [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_4884_floor__exists,axiom,
    ! [X2: rat] :
    ? [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X2))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_4885_floor__exists1,axiom,
    ! [X2: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y5: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y5)),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int)))) )
         => ( Y5 = X4 ) ) ) ).

% floor_exists1
tff(fact_4886_floor__exists1,axiom,
    ! [X2: rat] :
    ? [X4: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),X2))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y5: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y5)),X2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int)))) )
         => ( Y5 = X4 ) ) ) ).

% floor_exists1
tff(fact_4887_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real))) ).

% of_int_ceiling_le_add_one
tff(fact_4888_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_4889_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_4890_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_4891_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X2)) ) ).

% of_nat_less_of_int_iff
tff(fact_4892_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X2)) ) ).

% of_nat_less_of_int_iff
tff(fact_4893_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X2)) ) ).

% of_nat_less_of_int_iff
tff(fact_4894_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_4895_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_4896_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_4897_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_4898_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_4899_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_4900_Suc__as__int,axiom,
    ! [X6: nat] : aa_nat_nat(suc,X6) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X6)),one_one_int)) ).

% Suc_as_int
tff(fact_4901_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_4902_nat__diff__distrib_H,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y3)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4903_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_4904_nat__div__distrib,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y3)) ) ) ).

% nat_div_distrib
tff(fact_4905_nat__div__distrib_H,axiom,
    ! [Y3: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y3)) ) ) ).

% nat_div_distrib'
tff(fact_4906_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_4907_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real))) ) ).

% int_le_real_less
tff(fact_4908_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).

% int_less_real_le
tff(fact_4909_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_4910_ceiling__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I4))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% ceiling_split
tff(fact_4911_ceiling__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim2889992004027027881ng_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I4))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% ceiling_split
tff(fact_4912_ceiling__eq__iff,axiom,
    ! [X2: real,A2: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X2) = A2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A2)),one_one_real)),X2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,A2))) ) ) ).

% ceiling_eq_iff
tff(fact_4913_ceiling__eq__iff,axiom,
    ! [X2: rat,A2: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X2) = A2 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A2)),one_one_rat)),X2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,A2))) ) ) ).

% ceiling_eq_iff
tff(fact_4914_ceiling__unique,axiom,
    ! [Z: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,Z)))
       => ( aa_real_int(archim7802044766580827645g_real,X2) = Z ) ) ) ).

% ceiling_unique
tff(fact_4915_ceiling__unique,axiom,
    ! [Z: int,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,Z)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X2) = Z ) ) ) ).

% ceiling_unique
tff(fact_4916_ceiling__correct,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X2))),one_one_real)),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X2)))) ) ).

% ceiling_correct
tff(fact_4917_ceiling__correct,axiom,
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X2))),one_one_rat)),X2))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X2)))) ) ).

% ceiling_correct
tff(fact_4918_ceiling__less__iff,axiom,
    ! [X2: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X2)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% ceiling_less_iff
tff(fact_4919_ceiling__less__iff,axiom,
    ! [X2: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_4920_le__ceiling__iff,axiom,
    ! [Z: int,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X2)) ) ).

% le_ceiling_iff
tff(fact_4921_le__ceiling__iff,axiom,
    ! [Z: int,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X2)) ) ).

% le_ceiling_iff
tff(fact_4922_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_4923_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_4924_real__of__int__div2,axiom,
    ! [N: int,X2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X2))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X2))))) ).

% real_of_int_div2
tff(fact_4925_real__of__int__div3,axiom,
    ! [N: int,X2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X2))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X2)))),one_one_real)) ).

% real_of_int_div3
tff(fact_4926_diff__nat__eq__if,axiom,
    ! [Z5: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_4927_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_4928_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_4929_ceiling__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),one_one_real)),Q3)),P2)) ) ).

% ceiling_divide_lower
tff(fact_4930_ceiling__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),one_one_rat)),Q3)),P2)) ) ).

% ceiling_divide_lower
tff(fact_4931_ceiling__eq,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim7802044766580827645g_real,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4932_ceiling__eq,axiom,
    ! [N: int,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,N)),one_one_rat)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4933_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,A2)),aa_nat_rat(semiri681578069525770553at_rat,B2)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_4934_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A2)),aa_nat_real(semiri5074537144036343181t_real,B2)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_4935_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_4936_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A2)),aa_nat_nat(semiri1316708129612266289at_nat,B2)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_4937_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,K) != zero_zero_complex )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4938_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4939_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4940_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4941_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4942_vebt__delete_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),B3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fFalse) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                 => ! [N3: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                       => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y3 = 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um_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4943_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3),zero_zero_nat)) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2) )
               => ! [N3: nat] :
                    ( ( Xa = aa_nat_nat(suc,N3) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y3 = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_4944_vebt__insert_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),B3) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fTrue) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
               => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3),Xa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
                 => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3),Xa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( Y3 = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4945_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                 => ! [Va3: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2),Xa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Xa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                       => ( ( Y3 = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y3 = one_one_nat ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_4946_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3),Xa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3),Xa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_4947_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_4948_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_4949_vebt__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa))
      <=> pp(Y3) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( pp(Y3)
                <=> ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ~ pp(Y3)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ pp(Y3)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ~ pp(Y3)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( pp(Y3)
                        <=> ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4950_vebt__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa))
               => ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa))
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4951_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa))
      <=> pp(Y3) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( pp(Y3)
                <=> ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ( ~ pp(Y3)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( ( pp(Y3)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3),Xa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4952_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa))
               => ~ ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3),Xa))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4953_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa))
               => ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3),Xa))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4954_vebt__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa))
               => ~ ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa))
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4955_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),Xa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),Xa))
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa))
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4956_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa))
      <=> pp(Y3) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ~ pp(Y3)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),Xa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ pp(Y3)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2) )
                 => ( ( pp(Y3)
                    <=> ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),Xa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( ( pp(Y3)
                      <=> ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( ( pp(Y3)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4957_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2),Vb2),Xa))
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa))
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4958_set__bit__0,axiom,
    ! [A2: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4959_set__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4960_pred__less__length__list,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_4961_pred__lesseq__max,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% pred_lesseq_max
tff(fact_4962_add__shift,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X2)),some_nat(Y3)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_4963_mul__shift,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y3) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X2)),some_nat(Y3)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_4964_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_4965_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_4966_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4967_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_4968_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X2))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% succ_greatereq_min
tff(fact_4969_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K))) ).

% set_bit_greater_eq
tff(fact_4970_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),X2) = some_nat(Ma) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),X2) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_4971_vebt__succ_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),X2) = some_nat(Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary),X2) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4972_vebt__pred_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_pred(X2,Xa) = Y3 )
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != none_nat ) ) )
       => ( ! [A3: bool] :
              ( ? [Uw2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)
             => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A3)
                     => ( Y3 = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y3 = none_nat ) ) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
               => ( ? [Va3: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va3))
                 => ~ ( ( pp(B3)
                       => ( Y3 = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( ( pp(A3)
                           => ( Y3 = some_nat(zero_zero_nat) ) )
                          & ( ~ pp(A3)
                           => ( Y3 = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2)
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y3 != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)
                   => ( Y3 != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y3 = some_nat(Ma2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y3 = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_4973_vebt__succ_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_succ(X2,Xa) = Y3 )
     => ( ! [Uu2: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( pp(B3)
                   => ( Y3 = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( Y3 = none_nat ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)
           => ( ? [N3: nat] : Xa = aa_nat_nat(suc,N3)
             => ( Y3 != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y3 != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)
                 => ( Y3 != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y3 = some_nat(Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y3 = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_4974_vebt__succ_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_succ(X2,Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( pp(B3)
                     => ( Y3 = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y3 = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3),zero_zero_nat)) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2) )
               => ! [N3: nat] :
                    ( ( Xa = aa_nat_nat(suc,N3) )
                   => ( ( Y3 = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y3 = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y3 = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y3 = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y3 = some_nat(Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y3 = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_4975_vebt__pred_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: option_nat] :
      ( ( vEBT_vebt_pred(X2,Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( ( pp(A3)
                       => ( Y3 = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y3 = none_nat ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                 => ! [Va3: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                     => ( ( ( pp(B3)
                           => ( Y3 = some_nat(one_one_nat) ) )
                          & ( ~ pp(B3)
                           => ( ( pp(A3)
                               => ( Y3 = some_nat(zero_zero_nat) ) )
                              & ( ~ pp(A3)
                               => ( Y3 = none_nat ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2) )
                   => ( ( Y3 = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2),Xa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y3 = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Xa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                       => ( ( Y3 = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y3 = some_nat(Ma2) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y3 = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4976_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% succ_bound_size_univ
tff(fact_4977_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% insert_bound_size_univ
tff(fact_4978_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% pred_bound_size_univ
tff(fact_4979_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)) = minus_minus_set_nat(vEBT_set_vebt(T),insert_nat(X2,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_4980_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X2)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(T),insert_nat(X2,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_4981_insert__subset,axiom,
    ! [X2: nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(insert_nat(X2,A)),B))
    <=> ( pp(aa_set_nat_bool(member_nat(X2),B))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% insert_subset
tff(fact_4982_insert__subset,axiom,
    ! [X2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(X2,A)),B))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B))
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B)) ) ) ).

% insert_subset
tff(fact_4983_insert__subset,axiom,
    ! [X2: real,A: set_real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X2,A)),B))
    <=> ( pp(aa_set_real_bool(member_real(X2),B))
        & pp(aa_set_real_bool(ord_less_eq_set_real(A),B)) ) ) ).

% insert_subset
tff(fact_4984_insert__subset,axiom,
    ! [X2: int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X2,A)),B))
    <=> ( pp(aa_set_int_bool(member_int(X2),B))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% insert_subset
tff(fact_4985_Diff__insert0,axiom,
    ! [X2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
     => ( minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,B)) = minus_5127226145743854075T_VEBT(A,B) ) ) ).

% Diff_insert0
tff(fact_4986_Diff__insert0,axiom,
    ! [X2: real,A: set_real,B: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A))
     => ( minus_minus_set_real(A,insert_real(X2,B)) = minus_minus_set_real(A,B) ) ) ).

% Diff_insert0
tff(fact_4987_Diff__insert0,axiom,
    ! [X2: int,A: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A))
     => ( minus_minus_set_int(A,insert_int(X2,B)) = minus_minus_set_int(A,B) ) ) ).

% Diff_insert0
tff(fact_4988_Diff__insert0,axiom,
    ! [X2: nat,A: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
     => ( minus_minus_set_nat(A,insert_nat(X2,B)) = minus_minus_set_nat(A,B) ) ) ).

% Diff_insert0
tff(fact_4989_insert__Diff1,axiom,
    ! [X2: vEBT_VEBT,B: set_VEBT_VEBT,A: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B))
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A),B) = minus_5127226145743854075T_VEBT(A,B) ) ) ).

% insert_Diff1
tff(fact_4990_insert__Diff1,axiom,
    ! [X2: real,B: set_real,A: set_real] :
      ( pp(aa_set_real_bool(member_real(X2),B))
     => ( minus_minus_set_real(insert_real(X2,A),B) = minus_minus_set_real(A,B) ) ) ).

% insert_Diff1
tff(fact_4991_insert__Diff1,axiom,
    ! [X2: int,B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(member_int(X2),B))
     => ( minus_minus_set_int(insert_int(X2,A),B) = minus_minus_set_int(A,B) ) ) ).

% insert_Diff1
tff(fact_4992_insert__Diff1,axiom,
    ! [X2: nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),B))
     => ( minus_minus_set_nat(insert_nat(X2,A),B) = minus_minus_set_nat(A,B) ) ) ).

% insert_Diff1
tff(fact_4993_singleton__conv2,axiom,
    ! [A2: vEBT_VEBT] : collect_VEBT_VEBT(fequal_VEBT_VEBT(A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_4994_singleton__conv2,axiom,
    ! [A2: complex] : collect_complex(aa_com1553036987509291606x_bool(fequal_complex,A2)) = insert_complex(A2,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_4995_singleton__conv2,axiom,
    ! [A2: product_prod_int_int] : collec213857154873943460nt_int(fequal6010348034681699726nt_int(A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ).

% singleton_conv2
tff(fact_4996_singleton__conv2,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_bool(fequal_nat,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_4997_singleton__conv2,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_bool(fequal_int,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv2
tff(fact_4998_singleton__conv,axiom,
    ! [A2: vEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_bx(A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_4999_singleton__conv,axiom,
    ! [A2: complex] : collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_by,A2)) = insert_complex(A2,bot_bot_set_complex) ).

% singleton_conv
tff(fact_5000_singleton__conv,axiom,
    ! [A2: product_prod_int_int] : collec213857154873943460nt_int(aTP_Lamm_bz(A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ).

% singleton_conv
tff(fact_5001_singleton__conv,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ca,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv
tff(fact_5002_singleton__conv,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_cb,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv
tff(fact_5003_singleton__insert__inj__eq,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(A2,A) )
    <=> ( ( A2 = B2 )
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5004_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( insert_nat(B2,bot_bot_set_nat) = insert_nat(A2,A) )
    <=> ( ( A2 = B2 )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(B2,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5005_singleton__insert__inj__eq,axiom,
    ! [B2: int,A2: int,A: set_int] :
      ( ( insert_int(B2,bot_bot_set_int) = insert_int(A2,A) )
    <=> ( ( A2 = B2 )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(B2,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5006_singleton__insert__inj__eq_H,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A2,A) = insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT) )
    <=> ( ( A2 = B2 )
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5007_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( insert_nat(A2,A) = insert_nat(B2,bot_bot_set_nat) )
    <=> ( ( A2 = B2 )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(B2,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5008_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B2: int] :
      ( ( insert_int(A2,A) = insert_int(B2,bot_bot_set_int) )
    <=> ( ( A2 = B2 )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(B2,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5009_insert__Diff__single,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] : insert_VEBT_VEBT(A2,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = insert_VEBT_VEBT(A2,A) ).

% insert_Diff_single
tff(fact_5010_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] : insert_int(A2,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = insert_int(A2,A) ).

% insert_Diff_single
tff(fact_5011_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] : insert_nat(A2,minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = insert_nat(A2,A) ).

% insert_Diff_single
tff(fact_5012_insert__compr,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT] : insert_VEBT_VEBT(A2,B) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_cc(A2),B)) ).

% insert_compr
tff(fact_5013_insert__compr,axiom,
    ! [A2: real,B: set_real] : insert_real(A2,B) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_cd(A2),B)) ).

% insert_compr
tff(fact_5014_insert__compr,axiom,
    ! [A2: nat,B: set_nat] : insert_nat(A2,B) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ce(A2),B)) ).

% insert_compr
tff(fact_5015_insert__compr,axiom,
    ! [A2: int,B: set_int] : insert_int(A2,B) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_cf(A2),B)) ).

% insert_compr
tff(fact_5016_insert__compr,axiom,
    ! [A2: complex,B: set_complex] : insert_complex(A2,B) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_cg(A2),B)) ).

% insert_compr
tff(fact_5017_insert__compr,axiom,
    ! [A2: product_prod_int_int,B: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(A2,B) = collec213857154873943460nt_int(aa_set1381271197253378494t_bool(aTP_Lamm_ch(A2),B)) ).

% insert_compr
tff(fact_5018_insert__Collect,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_bool] : insert_VEBT_VEBT(A2,collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_ci(A2),P)) ).

% insert_Collect
tff(fact_5019_insert__Collect,axiom,
    ! [A2: nat,P: fun_nat_bool] : insert_nat(A2,collect_nat(P)) = collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_cj(A2),P)) ).

% insert_Collect
tff(fact_5020_insert__Collect,axiom,
    ! [A2: int,P: fun_int_bool] : insert_int(A2,collect_int(P)) = collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ck(A2),P)) ).

% insert_Collect
tff(fact_5021_insert__Collect,axiom,
    ! [A2: complex,P: fun_complex_bool] : insert_complex(A2,collect_complex(P)) = collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_cl(A2),P)) ).

% insert_Collect
tff(fact_5022_insert__Collect,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr803113274115125473t_bool] : insert5033312907999012233nt_int(A2,collec213857154873943460nt_int(P)) = collec213857154873943460nt_int(aa_fun8917503761599146496t_bool(aTP_Lamm_cm(A2),P)) ).

% insert_Collect
tff(fact_5023_insert__mono,axiom,
    ! [C: set_nat,D: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(insert_nat(A2,C)),insert_nat(A2,D))) ) ).

% insert_mono
tff(fact_5024_insert__mono,axiom,
    ! [C: set_VEBT_VEBT,D: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(C),D))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(A2,C)),insert_VEBT_VEBT(A2,D))) ) ).

% insert_mono
tff(fact_5025_insert__mono,axiom,
    ! [C: set_int,D: set_int,A2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),D))
     => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(A2,C)),insert_int(A2,D))) ) ).

% insert_mono
tff(fact_5026_subset__insert,axiom,
    ! [X2: nat,A: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(X2,B)))
      <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% subset_insert
tff(fact_5027_subset__insert,axiom,
    ! [X2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(X2,B)))
      <=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B)) ) ) ).

% subset_insert
tff(fact_5028_subset__insert,axiom,
    ! [X2: real,A: set_real,B: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),insert_real(X2,B)))
      <=> pp(aa_set_real_bool(ord_less_eq_set_real(A),B)) ) ) ).

% subset_insert
tff(fact_5029_subset__insert,axiom,
    ! [X2: int,A: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(X2,B)))
      <=> pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% subset_insert
tff(fact_5030_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(B),insert_nat(A2,B))) ).

% subset_insertI
tff(fact_5031_subset__insertI,axiom,
    ! [B: set_VEBT_VEBT,A2: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),insert_VEBT_VEBT(A2,B))) ).

% subset_insertI
tff(fact_5032_subset__insertI,axiom,
    ! [B: set_int,A2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(B),insert_int(A2,B))) ).

% subset_insertI
tff(fact_5033_subset__insertI2,axiom,
    ! [A: set_nat,B: set_nat,B2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(B2,B))) ) ).

% subset_insertI2
tff(fact_5034_subset__insertI2,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B2,B))) ) ).

% subset_insertI2
tff(fact_5035_subset__insertI2,axiom,
    ! [A: set_int,B: set_int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(B2,B))) ) ).

% subset_insertI2
tff(fact_5036_insert__Diff__if,axiom,
    ! [X2: vEBT_VEBT,B: set_VEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B))
       => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A),B) = minus_5127226145743854075T_VEBT(A,B) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B))
       => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A),B) = insert_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_5037_insert__Diff__if,axiom,
    ! [X2: real,B: set_real,A: set_real] :
      ( ( pp(aa_set_real_bool(member_real(X2),B))
       => ( minus_minus_set_real(insert_real(X2,A),B) = minus_minus_set_real(A,B) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X2),B))
       => ( minus_minus_set_real(insert_real(X2,A),B) = insert_real(X2,minus_minus_set_real(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_5038_insert__Diff__if,axiom,
    ! [X2: int,B: set_int,A: set_int] :
      ( ( pp(aa_set_int_bool(member_int(X2),B))
       => ( minus_minus_set_int(insert_int(X2,A),B) = minus_minus_set_int(A,B) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X2),B))
       => ( minus_minus_set_int(insert_int(X2,A),B) = insert_int(X2,minus_minus_set_int(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_5039_insert__Diff__if,axiom,
    ! [X2: nat,B: set_nat,A: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X2),B))
       => ( minus_minus_set_nat(insert_nat(X2,A),B) = minus_minus_set_nat(A,B) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X2),B))
       => ( minus_minus_set_nat(insert_nat(X2,A),B) = insert_nat(X2,minus_minus_set_nat(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_5040_Collect__conv__if2,axiom,
    ! [P: fun_VEBT_VEBT_bool,A2: vEBT_VEBT] :
      ( ( pp(aa_VEBT_VEBT_bool(P,A2))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_cn(P),A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ) )
      & ( ~ pp(aa_VEBT_VEBT_bool(P,A2))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_cn(P),A2)) = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
tff(fact_5041_Collect__conv__if2,axiom,
    ! [P: fun_complex_bool,A2: complex] :
      ( ( pp(aa_complex_bool(P,A2))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_co(P),A2)) = insert_complex(A2,bot_bot_set_complex) ) )
      & ( ~ pp(aa_complex_bool(P,A2))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_co(P),A2)) = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
tff(fact_5042_Collect__conv__if2,axiom,
    ! [P: fun_Pr803113274115125473t_bool,A2: product_prod_int_int] :
      ( ( pp(aa_Pro4501881432599481050t_bool(P,A2))
       => ( collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(aTP_Lamm_cp(P),A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ) )
      & ( ~ pp(aa_Pro4501881432599481050t_bool(P,A2))
       => ( collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(aTP_Lamm_cp(P),A2)) = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if2
tff(fact_5043_Collect__conv__if2,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ( pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cq(P),A2)) = insert_nat(A2,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cq(P),A2)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_5044_Collect__conv__if2,axiom,
    ! [P: fun_int_bool,A2: int] :
      ( ( pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_cr(P),A2)) = insert_int(A2,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_cr(P),A2)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_5045_Collect__conv__if,axiom,
    ! [P: fun_VEBT_VEBT_bool,A2: vEBT_VEBT] :
      ( ( pp(aa_VEBT_VEBT_bool(P,A2))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_cs(P),A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ) )
      & ( ~ pp(aa_VEBT_VEBT_bool(P,A2))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_cs(P),A2)) = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
tff(fact_5046_Collect__conv__if,axiom,
    ! [P: fun_complex_bool,A2: complex] :
      ( ( pp(aa_complex_bool(P,A2))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ct(P),A2)) = insert_complex(A2,bot_bot_set_complex) ) )
      & ( ~ pp(aa_complex_bool(P,A2))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ct(P),A2)) = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
tff(fact_5047_Collect__conv__if,axiom,
    ! [P: fun_Pr803113274115125473t_bool,A2: product_prod_int_int] :
      ( ( pp(aa_Pro4501881432599481050t_bool(P,A2))
       => ( collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(aTP_Lamm_cu(P),A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ) )
      & ( ~ pp(aa_Pro4501881432599481050t_bool(P,A2))
       => ( collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(aTP_Lamm_cu(P),A2)) = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if
tff(fact_5048_Collect__conv__if,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ( pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cv(P),A2)) = insert_nat(A2,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A2))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cv(P),A2)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_5049_Collect__conv__if,axiom,
    ! [P: fun_int_bool,A2: int] :
      ( ( pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_cw(P),A2)) = insert_int(A2,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A2))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_cw(P),A2)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_5050_subset__singletonD,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))
     => ( ( A = bot_bo8194388402131092736T_VEBT )
        | ( A = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_5051_subset__singletonD,axiom,
    ! [A: set_nat,X2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(X2,bot_bot_set_nat)))
     => ( ( A = bot_bot_set_nat )
        | ( A = insert_nat(X2,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_5052_subset__singletonD,axiom,
    ! [A: set_int,X2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(X2,bot_bot_set_int)))
     => ( ( A = bot_bot_set_int )
        | ( A = insert_int(X2,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_5053_subset__singleton__iff,axiom,
    ! [X: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(X),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))
    <=> ( ( X = bot_bo8194388402131092736T_VEBT )
        | ( X = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_5054_subset__singleton__iff,axiom,
    ! [X: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),insert_nat(A2,bot_bot_set_nat)))
    <=> ( ( X = bot_bot_set_nat )
        | ( X = insert_nat(A2,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_5055_subset__singleton__iff,axiom,
    ! [X: set_int,A2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),insert_int(A2,bot_bot_set_int)))
    <=> ( ( X = bot_bot_set_int )
        | ( X = insert_int(A2,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_5056_Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,B)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A,B),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)) ).

% Diff_insert
tff(fact_5057_Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] : minus_minus_set_int(A,insert_int(A2,B)) = minus_minus_set_int(minus_minus_set_int(A,B),insert_int(A2,bot_bot_set_int)) ).

% Diff_insert
tff(fact_5058_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : minus_minus_set_nat(A,insert_nat(A2,B)) = minus_minus_set_nat(minus_minus_set_nat(A,B),insert_nat(A2,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_5059_insert__Diff,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
     => ( insert_VEBT_VEBT(A2,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = A ) ) ).

% insert_Diff
tff(fact_5060_insert__Diff,axiom,
    ! [A2: real,A: set_real] :
      ( pp(aa_set_real_bool(member_real(A2),A))
     => ( insert_real(A2,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = A ) ) ).

% insert_Diff
tff(fact_5061_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( pp(aa_set_int_bool(member_int(A2),A))
     => ( insert_int(A2,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = A ) ) ).

% insert_Diff
tff(fact_5062_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A2),A))
     => ( insert_nat(A2,minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = A ) ) ).

% insert_Diff
tff(fact_5063_Diff__insert2,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,B)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)),B) ).

% Diff_insert2
tff(fact_5064_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B: set_int] : minus_minus_set_int(A,insert_int(A2,B)) = minus_minus_set_int(minus_minus_set_int(A,insert_int(A2,bot_bot_set_int)),B) ).

% Diff_insert2
tff(fact_5065_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : minus_minus_set_nat(A,insert_nat(A2,B)) = minus_minus_set_nat(minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat)),B) ).

% Diff_insert2
tff(fact_5066_Diff__insert__absorb,axiom,
    ! [X2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A),insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)) = A ) ) ).

% Diff_insert_absorb
tff(fact_5067_Diff__insert__absorb,axiom,
    ! [X2: real,A: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A))
     => ( minus_minus_set_real(insert_real(X2,A),insert_real(X2,bot_bot_set_real)) = A ) ) ).

% Diff_insert_absorb
tff(fact_5068_Diff__insert__absorb,axiom,
    ! [X2: int,A: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A))
     => ( minus_minus_set_int(insert_int(X2,A),insert_int(X2,bot_bot_set_int)) = A ) ) ).

% Diff_insert_absorb
tff(fact_5069_Diff__insert__absorb,axiom,
    ! [X2: nat,A: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
     => ( minus_minus_set_nat(insert_nat(X2,A),insert_nat(X2,bot_bot_set_nat)) = A ) ) ).

% Diff_insert_absorb
tff(fact_5070_set__minus__singleton__eq,axiom,
    ! [X2: vEBT_VEBT,X: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),X))
     => ( minus_5127226145743854075T_VEBT(X,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_5071_set__minus__singleton__eq,axiom,
    ! [X2: real,X: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),X))
     => ( minus_minus_set_real(X,insert_real(X2,bot_bot_set_real)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_5072_set__minus__singleton__eq,axiom,
    ! [X2: int,X: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),X))
     => ( minus_minus_set_int(X,insert_int(X2,bot_bot_set_int)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_5073_set__minus__singleton__eq,axiom,
    ! [X2: nat,X: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),X))
     => ( minus_minus_set_nat(X,insert_nat(X2,bot_bot_set_nat)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_5074_insert__minus__eq,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( X2 != Y3 )
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A),insert_VEBT_VEBT(Y3,bot_bo8194388402131092736T_VEBT)) = insert_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Y3,bot_bo8194388402131092736T_VEBT))) ) ) ).

% insert_minus_eq
tff(fact_5075_insert__minus__eq,axiom,
    ! [X2: int,Y3: int,A: set_int] :
      ( ( X2 != Y3 )
     => ( minus_minus_set_int(insert_int(X2,A),insert_int(Y3,bot_bot_set_int)) = insert_int(X2,minus_minus_set_int(A,insert_int(Y3,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_5076_insert__minus__eq,axiom,
    ! [X2: nat,Y3: nat,A: set_nat] :
      ( ( X2 != Y3 )
     => ( minus_minus_set_nat(insert_nat(X2,A),insert_nat(Y3,bot_bot_set_nat)) = insert_nat(X2,minus_minus_set_nat(A,insert_nat(Y3,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_5077_subset__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,X2: vEBT_VEBT,C: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),minus_5127226145743854075T_VEBT(B,insert_VEBT_VEBT(X2,C))))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),minus_5127226145743854075T_VEBT(B,C)))
        & ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A)) ) ) ).

% subset_Diff_insert
tff(fact_5078_subset__Diff__insert,axiom,
    ! [A: set_real,B: set_real,X2: real,C: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),minus_minus_set_real(B,insert_real(X2,C))))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A),minus_minus_set_real(B,C)))
        & ~ pp(aa_set_real_bool(member_real(X2),A)) ) ) ).

% subset_Diff_insert
tff(fact_5079_subset__Diff__insert,axiom,
    ! [A: set_nat,B: set_nat,X2: nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),minus_minus_set_nat(B,insert_nat(X2,C))))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),minus_minus_set_nat(B,C)))
        & ~ pp(aa_set_nat_bool(member_nat(X2),A)) ) ) ).

% subset_Diff_insert
tff(fact_5080_subset__Diff__insert,axiom,
    ! [A: set_int,B: set_int,X2: int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),minus_minus_set_int(B,insert_int(X2,C))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),minus_minus_set_int(B,C)))
        & ~ pp(aa_set_int_bool(member_int(X2),A)) ) ) ).

% subset_Diff_insert
tff(fact_5081_Diff__single__insert,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT))),B))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(X2,B))) ) ).

% Diff_single_insert
tff(fact_5082_Diff__single__insert,axiom,
    ! [A: set_nat,X2: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A,insert_nat(X2,bot_bot_set_nat))),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(X2,B))) ) ).

% Diff_single_insert
tff(fact_5083_Diff__single__insert,axiom,
    ! [A: set_int,X2: int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A,insert_int(X2,bot_bot_set_int))),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(X2,B))) ) ).

% Diff_single_insert
tff(fact_5084_subset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(X2,B)))
    <=> ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT))),B)) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B)) ) ) ) ).

% subset_insert_iff
tff(fact_5085_subset__insert__iff,axiom,
    ! [A: set_real,X2: real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),insert_real(X2,B)))
    <=> ( ( pp(aa_set_real_bool(member_real(X2),A))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A,insert_real(X2,bot_bot_set_real))),B)) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A),B)) ) ) ) ).

% subset_insert_iff
tff(fact_5086_subset__insert__iff,axiom,
    ! [A: set_nat,X2: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),insert_nat(X2,B)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X2),A))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A,insert_nat(X2,bot_bot_set_nat))),B)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ) ).

% subset_insert_iff
tff(fact_5087_subset__insert__iff,axiom,
    ! [A: set_int,X2: int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),insert_int(X2,B)))
    <=> ( ( pp(aa_set_int_bool(member_int(X2),A))
         => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A,insert_int(X2,bot_bot_set_int))),B)) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ) ).

% subset_insert_iff
tff(fact_5088_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% atLeast0_atMost_Suc
tff(fact_5089_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( set_or1269000886237332187st_nat(M,N) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5090_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5091_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = set_or1269000886237332187st_nat(M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_5092_remove__subset,axiom,
    ! [X2: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),S))
     => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT))),S)) ) ).

% remove_subset
tff(fact_5093_remove__subset,axiom,
    ! [X2: real,S: set_real] :
      ( pp(aa_set_real_bool(member_real(X2),S))
     => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(S,insert_real(X2,bot_bot_set_real))),S)) ) ).

% remove_subset
tff(fact_5094_remove__subset,axiom,
    ! [X2: int,S: set_int] :
      ( pp(aa_set_int_bool(member_int(X2),S))
     => pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(S,insert_int(X2,bot_bot_set_int))),S)) ) ).

% remove_subset
tff(fact_5095_remove__subset,axiom,
    ! [X2: nat,S: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),S))
     => pp(aa_set_nat_bool(ord_less_set_nat(minus_minus_set_nat(S,insert_nat(X2,bot_bot_set_nat))),S)) ) ).

% remove_subset
tff(fact_5096_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,X2)),insert_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_5097_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X2))),insert_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs)))) ).

% set_update_subset_insert
tff(fact_5098_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X2))),insert_nat(X2,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_5099_set__update__subset__insert,axiom,
    ! [Xs: list_bool,I: nat,X2: bool] : ord_less_eq_set_bool(set_bool2(list_update_bool(Xs,I,X2)),insert_bool(X2,set_bool2(Xs))) ).

% set_update_subset_insert
tff(fact_5100_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X2: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X2))),insert_real(X2,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_5101_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X2))),insert_int(X2,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_5102_psubset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A),insert_VEBT_VEBT(X2,B)))
    <=> ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B))
         => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A),B)) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),B))
         => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
             => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT))),B)) )
            & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
             => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_5103_psubset__insert__iff,axiom,
    ! [A: set_real,X2: real,B: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A),insert_real(X2,B)))
    <=> ( ( pp(aa_set_real_bool(member_real(X2),B))
         => pp(aa_set_real_bool(ord_less_set_real(A),B)) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),B))
         => ( ( pp(aa_set_real_bool(member_real(X2),A))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A,insert_real(X2,bot_bot_set_real))),B)) )
            & ( ~ pp(aa_set_real_bool(member_real(X2),A))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_5104_psubset__insert__iff,axiom,
    ! [A: set_nat,X2: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),insert_nat(X2,B)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X2),B))
         => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),B))
         => ( ( pp(aa_set_nat_bool(member_nat(X2),A))
             => pp(aa_set_nat_bool(ord_less_set_nat(minus_minus_set_nat(A,insert_nat(X2,bot_bot_set_nat))),B)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
             => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_5105_psubset__insert__iff,axiom,
    ! [A: set_int,X2: int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),insert_int(X2,B)))
    <=> ( ( pp(aa_set_int_bool(member_int(X2),B))
         => pp(aa_set_int_bool(ord_less_set_int(A),B)) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),B))
         => ( ( pp(aa_set_int_bool(member_int(X2),A))
             => pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(A,insert_int(X2,bot_bot_set_int))),B)) )
            & ( ~ pp(aa_set_int_bool(member_int(X2),A))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_5106_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S2)),X2) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_5107_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_5108_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_bool_VEBT_VEBT(vEBT_Leaf(Uv),Uw)),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_5109_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_5110_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_5111_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( insert_int(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,X2))) = insert_int(X2,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5112_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( insert_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,X2))) = insert_VEBT_VEBT(X2,set_VEBT_VEBT2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5113_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( insert_real(aa_nat_real(nth_real(L),I),set_real2(list_update_real(L,I,X2))) = insert_real(X2,set_real2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5114_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( insert_bool(aa_nat_bool(nth_bool(L),I),set_bool2(list_update_bool(L,I,X2))) = insert_bool(X2,set_bool2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5115_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( insert_nat(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,X2))) = insert_nat(X2,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5116_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( insert_VEBT_VEBTi(nth_VEBT_VEBTi(L,I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,X2))) = insert_VEBT_VEBTi(X2,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5117_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S2)),X2) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_5118_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,Va: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_5119_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_5120_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd3: list_VEBT_VEBT,Ve3: vEBT_VEBT,Vf2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd3),Ve3)),Vf2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_5121_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: bool,B2: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B2)),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_5122_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd3: vEBT_VEBT,Ve3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd3)),Ve3) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_5123_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),Uw)),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_5124_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_5125_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_5126_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),X2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_5127_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),Y3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ) ) ).

% insersimp
tff(fact_5128_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ) ) ).

% insertsimp
tff(fact_5129_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ).

% insert_bound_height
tff(fact_5130_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))) ) ).

% pred_bound_height
tff(fact_5131_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))) ) ).

% succ_bound_height
tff(fact_5132_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I6,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5133_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F3: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I6,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3),C2,Q)
           => hoare_2774773592644289642e_bool(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5134_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F3: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I6,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3),C2,Q)
           => hoare_7629718768684598413on_nat(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5135_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F3: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I6,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3),C2,Q)
           => hoare_3067605981109127869le_nat(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5136_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,X2),Xa) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)
               => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                   => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_5137_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),Mi),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_5138_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3),Xa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3),Xa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_5139_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,X2),Xa) = Y3 )
     => ( ( ? [Uu2: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)
           => ( ? [N3: nat] : Xa = aa_nat_nat(suc,N3)
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)
                 => ( Y3 != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                     => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_5140_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,X2),Xa) = Y3 )
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A3: bool,Uw2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
               => ( ? [Va3: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va3))
                 => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B3),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_5141_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,T)),one_one_nat)) ).

% minNull_bound
tff(fact_5142_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_5143_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_5144_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),fTrue)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_5145_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_5146_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5147_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_5148_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_5149_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_5150_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_5151_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_5152_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).

% maxt_bound
tff(fact_5153_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_5154_mint__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).

% mint_bound
tff(fact_5155_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A2),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_5156_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_5157_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X2) = Y3 )
     => ( ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
         => ( Y3 != one_one_nat ) )
       => ( ( ? [Uv2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
           => ( Y3 != one_one_nat ) )
         => ( ( ? [Uu2: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
             => ( Y3 != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ( Y3 != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2)
                 => ( Y3 != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_5158_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X2) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B3),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
             => ( Y3 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_5159_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X2) = Y3 )
     => ( ! [A3: bool] :
            ( ? [B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
           => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A3),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
             => ( Y3 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_5160_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one))))),one_one_nat))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus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VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),X2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_nat)))))),one_one_nat)))))) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_5161_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,X2),Xa) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( ( Xa = zero_zero_nat )
           => ( Y3 != one_one_nat ) ) )
       => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y3 != one_one_nat ) ) )
         => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
             => ( ? [N3: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N3))
               => ( Y3 != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)
               => ( Y3 != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2)
                 => ( Y3 != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)
                   => ( Y3 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one))))),one_one_nat))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_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) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_5162_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_5163_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_5164_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                 => ! [N3: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y3 = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                         => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one))))),one_one_nat))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))))),one_one_nat)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_nat)))))),one_one_nat)))))) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_5165_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B3),zero_zero_nat)) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2) )
               => ! [N3: nat] :
                    ( ( Xa = aa_nat_nat(suc,N3) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                       => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_5166_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,X2),Xa) = Y3 )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(X2,Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y3 = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
                 => ! [Va3: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                     => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B3),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va2),Xa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y3 = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Xa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                       => ( ( Y3 = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2) )
                         => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2),Summary2),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_5167_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X2: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( Xaa = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),X2)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xb),vEBT_vebt_mint(Xaa))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,H,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,H,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xb),vEBT_vebt_mint(Xaa))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,Xaa)),list_u6098035379799741383_VEBTi(Tree_is,H,X2))))) ) ) ).

% big_assn_simp'
tff(fact_5168_big__assn__simp,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,L: nat,X2: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)),X2)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xaa),vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,H,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,H,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xaa),vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),list_u6098035379799741383_VEBTi(Tree_is,H,X2))))) ) ).

% big_assn_simp
tff(fact_5169_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_VEBT_VEBT,Y3: vEBT_VEBT,X2: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_s6755466524823107622T_VEBT(TreeList4) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y3),X2)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2))))) ) ) ).

% tcd
tff(fact_5170_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_real,Y3: vEBT_VEBT,X2: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_real(TreeList4) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y3),X2)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2))))) ) ) ).

% tcd
tff(fact_5171_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_bool,Y3: vEBT_VEBT,X2: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_bool(TreeList4) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y3),X2)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2))))) ) ) ).

% tcd
tff(fact_5172_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_nat,Y3: vEBT_VEBT,X2: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_nat(TreeList4) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y3),X2)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2))))) ) ) ).

% tcd
tff(fact_5173_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_VEBT_VEBTi,Y3: vEBT_VEBT,X2: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_s7982070591426661849_VEBTi(TreeList4) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y3),X2)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),list_u6098035379799741383_VEBTi(Tree_is,I,X2))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y3)),list_u6098035379799741383_VEBTi(Tree_is,I,X2))))) ) ) ).

% tcd
tff(fact_5174_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or66887138388493659n_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5175_atLeastLessThan__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or8585797421378605585et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I))
        & pp(aa_set_int_bool(ord_less_set_int(I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5176_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or4029947393144176647an_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5177_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or1222409239386451017an_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5178_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),set_or4665077453230672383an_nat(L,U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5179_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or4662586982721622107an_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5180_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),set_or8404916559141939852nteger(L,U)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),I))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_5181_atLeastLessThan__empty,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( set_or8585797421378605585et_int(A2,B2) = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_5182_atLeastLessThan__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
     => ( set_or4029947393144176647an_rat(A2,B2) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_5183_atLeastLessThan__empty,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( set_or1222409239386451017an_num(A2,B2) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_5184_atLeastLessThan__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( set_or4665077453230672383an_nat(A2,B2) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_5185_atLeastLessThan__empty,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( set_or4662586982721622107an_int(A2,B2) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_5186_atLeastLessThan__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
     => ( set_or8404916559141939852nteger(A2,B2) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_5187_atLeastLessThan__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A2,B2) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5188_atLeastLessThan__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A2,B2) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5189_atLeastLessThan__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A2,B2) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5190_atLeastLessThan__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat = set_or4665077453230672383an_nat(A2,B2) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5191_atLeastLessThan__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A2,B2) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5192_atLeastLessThan__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A2,B2) )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5193_atLeastLessThan__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( set_or66887138388493659n_real(A2,B2) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5194_atLeastLessThan__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( set_or4029947393144176647an_rat(A2,B2) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5195_atLeastLessThan__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( set_or1222409239386451017an_num(A2,B2) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5196_atLeastLessThan__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( set_or4665077453230672383an_nat(A2,B2) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5197_atLeastLessThan__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( set_or4662586982721622107an_int(A2,B2) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5198_atLeastLessThan__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( set_or8404916559141939852nteger(A2,B2) = bot_bo3990330152332043303nteger )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5199_ivl__subset,axiom,
    ! [I: rat,J: rat,M: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(I,J)),set_or4029947393144176647an_rat(M,N)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,J),I))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M),I))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,J),N)) ) ) ) ).

% ivl_subset
tff(fact_5200_ivl__subset,axiom,
    ! [I: num,J: num,M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(I,J)),set_or1222409239386451017an_num(M,N)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,J),I))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),I))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,J),N)) ) ) ) ).

% ivl_subset
tff(fact_5201_ivl__subset,axiom,
    ! [I: nat,J: nat,M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_or4665077453230672383an_nat(I,J)),set_or4665077453230672383an_nat(M,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),N)) ) ) ) ).

% ivl_subset
tff(fact_5202_ivl__subset,axiom,
    ! [I: int,J: int,M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or4662586982721622107an_int(I,J)),set_or4662586982721622107an_int(M,N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),I))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),N)) ) ) ) ).

% ivl_subset
tff(fact_5203_ivl__subset,axiom,
    ! [I: code_integer,J: code_integer,M: code_integer,N: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I,J)),set_or8404916559141939852nteger(M,N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,J),I))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,M),I))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,J),N)) ) ) ) ).

% ivl_subset
tff(fact_5204_ivl__diff,axiom,
    ! [I: rat,N: rat,M: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),N))
     => ( minus_minus_set_rat(set_or4029947393144176647an_rat(I,M),set_or4029947393144176647an_rat(I,N)) = set_or4029947393144176647an_rat(N,M) ) ) ).

% ivl_diff
tff(fact_5205_ivl__diff,axiom,
    ! [I: num,N: num,M: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),N))
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I,M),set_or1222409239386451017an_num(I,N)) = set_or1222409239386451017an_num(N,M) ) ) ).

% ivl_diff
tff(fact_5206_ivl__diff,axiom,
    ! [I: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( minus_minus_set_nat(set_or4665077453230672383an_nat(I,M),set_or4665077453230672383an_nat(I,N)) = set_or4665077453230672383an_nat(N,M) ) ) ).

% ivl_diff
tff(fact_5207_ivl__diff,axiom,
    ! [I: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),N))
     => ( minus_minus_set_int(set_or4662586982721622107an_int(I,M),set_or4662586982721622107an_int(I,N)) = set_or4662586982721622107an_int(N,M) ) ) ).

% ivl_diff
tff(fact_5208_ivl__diff,axiom,
    ! [I: code_integer,N: code_integer,M: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),N))
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I,M),set_or8404916559141939852nteger(I,N)) = set_or8404916559141939852nteger(N,M) ) ) ).

% ivl_diff
tff(fact_5209_local_Oext,axiom,
    ! [Y3: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_s6755466524823107622T_VEBT(TreeList)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),Y3)),nth_VEBT_VEBTi(Tree_is,Y3))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y3,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y3,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),Y3)),nth_VEBT_VEBTi(Tree_is,Y3))))) ) ).

% local.ext
tff(fact_5210_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_5211_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I)),nth_VEBT_VEBTi(Tree_is,I))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is)))) ) ).

% recomp
tff(fact_5212_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I)),nth_VEBT_VEBTi(Tree_is,I))),Rest)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is)))) ) ).

% repack
tff(fact_5213_txe,axiom,
    ! [Y3: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_s6755466524823107622T_VEBT(TreeList)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),Y3)),nth_VEBT_VEBTi(Tree_is,Y3))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y3,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13),Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is)))) ) ).

% txe
tff(fact_5214_atLeastLessThan__eq__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
       => ( ( set_or66887138388493659n_real(A2,B2) = set_or66887138388493659n_real(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5215_atLeastLessThan__eq__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
       => ( ( set_or4029947393144176647an_rat(A2,B2) = set_or4029947393144176647an_rat(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5216_atLeastLessThan__eq__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),D2))
       => ( ( set_or1222409239386451017an_num(A2,B2) = set_or1222409239386451017an_num(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5217_atLeastLessThan__eq__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
       => ( ( set_or4665077453230672383an_nat(A2,B2) = set_or4665077453230672383an_nat(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5218_atLeastLessThan__eq__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
       => ( ( set_or4662586982721622107an_int(A2,B2) = set_or4662586982721622107an_int(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5219_atLeastLessThan__eq__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),D2))
       => ( ( set_or8404916559141939852nteger(A2,B2) = set_or8404916559141939852nteger(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5220_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( set_or66887138388493659n_real(A2,B2) = set_or66887138388493659n_real(C2,D2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5221_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A2,B2) = set_or4029947393144176647an_rat(C2,D2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5222_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A2,B2) = set_or1222409239386451017an_num(C2,D2) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),D2))
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5223_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( set_or4665077453230672383an_nat(A2,B2) = set_or4665077453230672383an_nat(C2,D2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5224_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( set_or4662586982721622107an_int(A2,B2) = set_or4662586982721622107an_int(C2,D2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5225_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A2,B2) = set_or8404916559141939852nteger(C2,D2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),D2))
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5226_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( set_or66887138388493659n_real(A2,B2) = set_or66887138388493659n_real(C2,D2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D2))
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5227_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A2,B2) = set_or4029947393144176647an_rat(C2,D2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D2))
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5228_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A2,B2) = set_or1222409239386451017an_num(C2,D2) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),D2))
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5229_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( set_or4665077453230672383an_nat(A2,B2) = set_or4665077453230672383an_nat(C2,D2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D2))
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5230_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( set_or4662586982721622107an_int(A2,B2) = set_or4662586982721622107an_int(C2,D2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D2))
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5231_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A2,B2) = set_or8404916559141939852nteger(C2,D2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),D2))
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5232_atLeastLessThan__subset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A2,B2)),set_or4029947393144176647an_rat(C2,D2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),D2)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5233_atLeastLessThan__subset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(A2,B2)),set_or1222409239386451017an_num(C2,D2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),D2)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5234_atLeastLessThan__subset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_or4665077453230672383an_nat(A2,B2)),set_or4665077453230672383an_nat(C2,D2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D2)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5235_atLeastLessThan__subset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or4662586982721622107an_int(A2,B2)),set_or4662586982721622107an_int(C2,D2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D2)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5236_atLeastLessThan__subset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A2,B2)),set_or8404916559141939852nteger(C2,D2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),D2)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5237_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N))
         => pp(aa_nat_bool(P,M7)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_5238_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N))
          & pp(aa_nat_bool(P,M7)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_5239_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_5240_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% atLeast0_lessThan_Suc
tff(fact_5241_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A2,B2)),set_or633870826150836451st_rat(C2,D2)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),D2)) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_5242_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or66887138388493659n_real(A2,B2)),set_or1222579329274155063t_real(C2,D2)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),D2)) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_5243_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int,D2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A2,B2),set_or8585797421378605585et_int(C2,D2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A2))
          & pp(aa_set_int_bool(ord_less_set_int(B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5244_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or4029947393144176647an_rat(C2,D2)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5245_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A2,B2)),set_or1222409239386451017an_num(C2,D2)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A2))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5246_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A2,B2)),set_or66887138388493659n_real(C2,D2)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),A2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5247_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_or1269000886237332187st_nat(A2,B2)),set_or4665077453230672383an_nat(C2,D2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5248_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A2,B2)),set_or4662586982721622107an_int(C2,D2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5249_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A2,B2)),set_or8404916559141939852nteger(C2,D2)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),D2)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5250_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: real,B2: real] : set_or66887138388493659n_real(A2,B2) = minus_minus_set_real(set_or1222579329274155063t_real(A2,B2),insert_real(B2,bot_bot_set_real)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5251_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: nat,B2: nat] : set_or4665077453230672383an_nat(A2,B2) = minus_minus_set_nat(set_or1269000886237332187st_nat(A2,B2),insert_nat(B2,bot_bot_set_nat)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5252_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: int,B2: int] : set_or4662586982721622107an_int(A2,B2) = minus_minus_set_int(set_or1266510415728281911st_int(A2,B2),insert_int(B2,bot_bot_set_int)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5253_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: code_integer,B2: code_integer] : set_or8404916559141939852nteger(A2,B2) = minus_2355218937544613996nteger(set_or189985376899183464nteger(A2,B2),insert_Code_integer(B2,bot_bo3990330152332043303nteger)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5254_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(M,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_5255_rule__at__index,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L1279224858307276611T_VEBT(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis7580683478039494528i_assn(aTP_Lamm_cx(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5256_rule__at__index,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A,Xs),Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis3335532474104912652i_assn(aTP_Lamm_cy(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5257_rule__at__index,axiom,
    ! [P: assn,A: fun_VE1323945509527881313l_assn,Xs: list_VEBT_VEBT,Xsi: list_bool,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_list_bool_assn(vEBT_L5320321625355440422T_bool(A,Xs),Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis3429412498766249874i_assn(aa_lis8505491871031121891i_assn(aTP_Lamm_cz(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5258_rule__at__index,axiom,
    ! [P: assn,A: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xsi: list_real,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L5781919052683127133T_real(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L4281036506115550016T_real(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L4281036506115550016T_real(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis1734048899804268331i_assn(aa_lis578560538171925480i_assn(aTP_Lamm_da(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5259_rule__at__index,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L4595930785310033027T_VEBT(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis9172383540912000132i_assn(aTP_Lamm_db(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5260_rule__at__index,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xs: list_real,Xsi: list_VEBT_VEBTi,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L9060850011106065574_VEBTi(A,Xs),Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis6960512757483956916i_assn(aa_lis4969729103256914541i_assn(aTP_Lamm_dc(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5261_rule__at__index,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L1446010312343316929al_nat(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis4680633809818909960i_assn(aTP_Lamm_dd(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5262_rule__at__index,axiom,
    ! [P: assn,A: fun_re2206215847901373695l_assn,Xs: list_real,Xsi: list_bool,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L870199270341830246l_bool(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis3429412498766249874i_assn(aa_lis5807736783900374247i_assn(aTP_Lamm_de(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5263_rule__at__index,axiom,
    ! [P: assn,A: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L1930518968523514909l_real(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L5184575500739366650l_real(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L5184575500739366650l_real(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis1734048899804268331i_assn(aa_lis8605243017695430628i_assn(aTP_Lamm_df(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5264_rule__at__index,axiom,
    ! [P: assn,A: fun_bo5394993412183903963T_assn,Xs: list_bool,Xsi: list_VEBT_VEBT,F3: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(vEBT_L2136499394512906784T_VEBT(A,Xs,Xsi)),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_bool(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3),C2,Q2)
         => ( ! [R3: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(Q2,R3)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_bool(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3))))
           => hoare_1429296392585015714_VEBTi(P,C2,aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis6766312878945579869i_assn(aTP_Lamm_dg(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5265_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,X2: vEBT_VEBT,Xi: vEBT_VEBT,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,X2),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5266_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,X2: vEBT_VEBT,Xi: nat,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,X2),Xi)),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_update_nat(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5267_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE1323945509527881313l_assn,X2: vEBT_VEBT,Xi: bool,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_bool,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,X2),Xi)),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L6176004506706552451T_bool(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_update_bool(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5268_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE6910866152645720856l_assn,X2: vEBT_VEBT,Xi: real,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_real,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,X2),Xi)),vEBT_L4281036506115550016T_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L4281036506115550016T_real(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_update_real(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5269_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,X2: real,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,X2),Xi)),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,list_update_real(Xs,I,X2)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5270_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,X2: real,Xi: vEBT_VEBT,I6: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,X2),Xi)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I6,A,list_update_real(Xs,I,X2),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5271_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,X2: real,Xi: nat,I6: set_nat,I: nat,Xs: list_real,Xsi: list_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,X2),Xi)),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I6,A,list_update_real(Xs,I,X2),list_update_nat(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5272_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re2206215847901373695l_assn,X2: real,Xi: bool,I6: set_nat,I: nat,Xs: list_real,Xsi: list_bool,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,X2),Xi)),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3606510757799916617l_bool(I6,A,list_update_real(Xs,I,X2),list_update_bool(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5273_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re6303261851654292474l_assn,X2: real,Xi: real,I6: set_nat,I: nat,Xs: list_real,Xsi: list_real,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,X2),Xi)),vEBT_L5184575500739366650l_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L5184575500739366650l_real(I6,A,list_update_real(Xs,I,X2),list_update_real(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5274_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_bo7535889243153246212i_assn,X2: bool,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBTi,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A,X2),Xi)),aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(I6,A,list_update_bool(Xs,I,X2)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5275_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5276_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5277_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_VE1323945509527881313l_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_bool,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5278_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_real,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L4281036506115550016T_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5279_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5280_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5281_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5282_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re2206215847901373695l_assn,Xs: list_real,I: nat,Xsi: list_bool,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5283_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re6303261851654292474l_assn,Xs: list_real,I: nat,Xsi: list_real,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L5184575500739366650l_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5284_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_bo5394993412183903963T_assn,Xs: list_bool,I: nat,Xsi: list_VEBT_VEBT,I6: set_nat,F3: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L2992182275864018813T_VEBT(I6,A,Xs,Xsi)),F3),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5285_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,X2: vEBT_VEBT,Xi: vEBT_VEBT,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,X2),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5286_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,X2: vEBT_VEBT,Xi: nat,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,X2),Xi)),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_update_nat(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5287_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE1323945509527881313l_assn,X2: vEBT_VEBT,Xi: bool,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_bool,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,X2),Xi)),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L6176004506706552451T_bool(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_update_bool(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5288_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE6910866152645720856l_assn,X2: vEBT_VEBT,Xi: real,I6: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_real,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,X2),Xi)),vEBT_L4281036506115550016T_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L4281036506115550016T_real(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X2),list_update_real(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5289_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,X2: real,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,X2),Xi)),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,list_update_real(Xs,I,X2)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5290_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,X2: real,Xi: vEBT_VEBT,I6: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,X2),Xi)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I6,A,list_update_real(Xs,I,X2),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5291_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,X2: real,Xi: nat,I6: set_nat,I: nat,Xs: list_real,Xsi: list_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,X2),Xi)),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I6,A,list_update_real(Xs,I,X2),list_update_nat(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5292_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re2206215847901373695l_assn,X2: real,Xi: bool,I6: set_nat,I: nat,Xs: list_real,Xsi: list_bool,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,X2),Xi)),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L3606510757799916617l_bool(I6,A,list_update_real(Xs,I,X2),list_update_bool(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5293_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re6303261851654292474l_assn,X2: real,Xi: real,I6: set_nat,I: nat,Xs: list_real,Xsi: list_real,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,X2),Xi)),vEBT_L5184575500739366650l_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L5184575500739366650l_real(I6,A,list_update_real(Xs,I,X2),list_update_real(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5294_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_bo7535889243153246212i_assn,X2: bool,Xi: vEBT_VEBTi,I6: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBTi,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A,X2),Xi)),aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(I6,A,list_update_bool(Xs,I,X2)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5295_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5296_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5297_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_VE1323945509527881313l_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_bool,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5298_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_real,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L4281036506115550016T_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5299_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5300_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5301_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5302_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re2206215847901373695l_assn,Xs: list_real,I: nat,Xsi: list_bool,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5303_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re6303261851654292474l_assn,Xs: list_real,I: nat,Xsi: list_real,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L5184575500739366650l_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5304_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_bo5394993412183903963T_assn,Xs: list_bool,I: nat,Xsi: list_VEBT_VEBT,I6: set_nat,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(vEBT_L2992182275864018813T_VEBT(I6,A,Xs,Xsi)),F3)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5305_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5306_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5307_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1323945509527881313l_assn,Xsi: list_bool] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L6176004506706552451T_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5308_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE6910866152645720856l_assn,Xsi: list_real] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L4281036506115550016T_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5309_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5310_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5311_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5312_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re2206215847901373695l_assn,Xsi: list_bool] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L3606510757799916617l_bool(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5313_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re6303261851654292474l_assn,Xsi: list_real] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L5184575500739366650l_real(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5314_listI__assn__extract,axiom,
    ! [I: nat,I6: set_nat,Xs: list_bool,A: fun_bo5394993412183903963T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( vEBT_L2992182275864018813T_VEBT(I6,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(I6,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5315_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_5316_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A: fun_int_fun_int_assn,A6: fun_int_fun_int_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: int] :
              ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs4)))
             => ( pp(aa_set_int_bool(member_int(Xi2),set_int2(Xsi2)))
               => ( aa_int_assn(aa_int_fun_int_assn(A,X4),Xi2) = aa_int_assn(aa_int_fun_int_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L74593716426352029nt_int(A,Xs,Xsi) = vEBT_L74593716426352029nt_int(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5317_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_in7755481285845205148T_assn,A6: fun_in7755481285845205148T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: vEBT_VEBT] :
              ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs4)))
             => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Xi2),set_VEBT_VEBT2(Xsi2)))
               => ( aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A,X4),Xi2) = aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L1664421287176695555T_VEBT(A,Xs,Xsi) = vEBT_L1664421287176695555T_VEBT(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5318_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real,A: fun_in6446958447386763770l_assn,A6: fun_in6446958447386763770l_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: real] :
              ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs4)))
             => ( pp(aa_set_real_bool(member_real(Xi2),set_real2(Xsi2)))
               => ( aa_real_assn(aa_int_fun_real_assn(A,X4),Xi2) = aa_real_assn(aa_int_fun_real_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L8288995350762215837t_real(A,Xs,Xsi) = vEBT_L8288995350762215837t_real(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5319_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_nat,Xsi2: list_nat,A: fun_int_fun_nat_assn,A6: fun_int_fun_nat_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: nat] :
              ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs4)))
             => ( pp(aa_set_nat_bool(member_nat(Xi2),set_nat2(Xsi2)))
               => ( aa_nat_assn(aa_int_fun_nat_assn(A,X4),Xi2) = aa_nat_assn(aa_int_fun_nat_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L77084186935402305nt_nat(A,Xs,Xsi) = vEBT_L77084186935402305nt_nat(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5320_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A: fun_VE123837653852366616t_assn,A6: fun_VE123837653852366616t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: vEBT_VEBT,Xi2: int] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs4)))
             => ( pp(aa_set_int_bool(member_int(Xi2),set_int2(Xsi2)))
               => ( aa_int_assn(aa_VEB392321844297299145t_assn(A,X4),Xi2) = aa_int_assn(aa_VEB392321844297299145t_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L8294436054247626077BT_int(A,Xs,Xsi) = vEBT_L8294436054247626077BT_int(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5321_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,A6: fun_VE1571388752190420990T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: vEBT_VEBT,Xi2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs4)))
             => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Xi2),set_VEBT_VEBT2(Xsi2)))
               => ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,X4),Xi2) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L1279224858307276611T_VEBT(A,Xs,Xsi) = vEBT_L1279224858307276611T_VEBT(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5322_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A: fun_VE6910866152645720856l_assn,A6: fun_VE6910866152645720856l_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: vEBT_VEBT,Xi2: real] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs4)))
             => ( pp(aa_set_real_bool(member_real(Xi2),set_real2(Xsi2)))
               => ( aa_real_assn(aa_VEB8575627266564790985l_assn(A,X4),Xi2) = aa_real_assn(aa_VEB8575627266564790985l_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L5781919052683127133T_real(A,Xs,Xsi) = vEBT_L5781919052683127133T_real(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5323_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A: fun_VE4931415397359990460t_assn,A6: fun_VE4931415397359990460t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: vEBT_VEBT,Xi2: nat] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs4)))
             => ( pp(aa_set_nat_bool(member_nat(Xi2),set_nat2(Xsi2)))
               => ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A,X4),Xi2) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A6,X4),Xi2) ) ) )
         => ( aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A,Xs),Xsi) = aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A6,Xs4),Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5324_list__assn__cong,axiom,
    ! [Xs: list_real,Xs4: list_real,Xsi: list_int,Xsi2: list_int,A: fun_re4990660043736809722t_assn,A6: fun_re4990660043736809722t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: real,Xi2: int] :
              ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs4)))
             => ( pp(aa_set_int_bool(member_int(Xi2),set_int2(Xsi2)))
               => ( aa_int_assn(aa_real_fun_int_assn(A,X4),Xi2) = aa_int_assn(aa_real_fun_int_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L1443519841834266653al_int(A,Xs,Xsi) = vEBT_L1443519841834266653al_int(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5325_list__assn__cong,axiom,
    ! [Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_re8050108611700907676T_assn,A6: fun_re8050108611700907676T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: real,Xi2: vEBT_VEBT] :
              ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs4)))
             => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Xi2),set_VEBT_VEBT2(Xsi2)))
               => ( aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,X4),Xi2) = aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A6,X4),Xi2) ) ) )
         => ( vEBT_L4595930785310033027T_VEBT(A,Xs,Xsi) = vEBT_L4595930785310033027T_VEBT(A6,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_5326_list__assn__mono,axiom,
    ! [P: fun_VE905575822219544865i_assn,P3: fun_VE905575822219544865i_assn,L: list_VEBT_VEBT,L3: list_VEBT_VEBTi] :
      ( ! [X4: vEBT_VEBT,X7: vEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,X4),X7)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P3,X4),X7)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,L),L3)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P3,L),L3))) ) ).

% list_assn_mono
tff(fact_5327_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L1279224858307276611T_VEBT(A,Xs,Xsi)),H))
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5328_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L4595930785310033027T_VEBT(A,Xs,Xsi)),H))
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5329_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_bo5394993412183903963T_assn,Xs: list_bool,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L2136499394512906784T_VEBT(A,Xs,Xsi)),H))
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_bool(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5330_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_na4489259242115493880T_assn,Xs: list_nat,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L8158188754432654943T_VEBT(A,Xs,Xsi)),H))
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5331_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L7265847600308530106T_VEBT(A,Xs,Xsi)),H))
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_s7982070591426661849_VEBTi(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5332_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L5781919052683127133T_real(A,Xs,Xsi)),H))
     => ( size_size_list_real(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5333_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L1930518968523514909l_real(A,Xs,Xsi)),H))
     => ( size_size_list_real(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5334_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_bo8885878543415060347l_assn,Xs: list_bool,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L5107238702033023424l_real(A,Xs,Xsi)),H))
     => ( size_size_list_real(Xsi) = size_size_list_bool(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5335_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_na448437756743965086l_assn,Xs: list_nat,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L6102073776069194049t_real(A,Xs,Xsi)),H))
     => ( size_size_list_real(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5336_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE1885522831824361987l_assn,Xs: list_VEBT_VEBTi,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(vEBT_L8937798142398754470i_real(A,Xs,Xsi)),H))
     => ( size_size_list_real(Xsi) = size_s7982070591426661849_VEBTi(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5337_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_s6755466524823107622T_VEBT(Li2) )
     => ( vEBT_L1279224858307276611T_VEBT(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5338_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_real,A: fun_VE6910866152645720856l_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_size_list_real(Li2) )
     => ( vEBT_L5781919052683127133T_real(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5339_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_bool,A: fun_VE1323945509527881313l_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_size_list_bool(Li2) )
     => ( aa_list_bool_assn(vEBT_L5320321625355440422T_bool(A,L),Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5340_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_nat,A: fun_VE4931415397359990460t_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_size_list_nat(Li2) )
     => ( aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A,L),Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5341_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_VEBT_VEBT,A: fun_re8050108611700907676T_assn] :
      ( ( size_size_list_real(L) != size_s6755466524823107622T_VEBT(Li2) )
     => ( vEBT_L4595930785310033027T_VEBT(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5342_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_real,A: fun_re6303261851654292474l_assn] :
      ( ( size_size_list_real(L) != size_size_list_real(Li2) )
     => ( vEBT_L1930518968523514909l_real(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5343_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_bool,A: fun_re2206215847901373695l_assn] :
      ( ( size_size_list_real(L) != size_size_list_bool(Li2) )
     => ( vEBT_L870199270341830246l_bool(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5344_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_nat,A: fun_re574865750389657758t_assn] :
      ( ( size_size_list_real(L) != size_size_list_nat(Li2) )
     => ( vEBT_L1446010312343316929al_nat(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5345_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_VEBT_VEBTi,A: fun_re560840401387337987i_assn] :
      ( ( size_size_list_real(L) != size_s7982070591426661849_VEBTi(Li2) )
     => ( aa_lis2508312872377092907i_assn(vEBT_L9060850011106065574_VEBTi(A,L),Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5346_list__assn__aux__ineq__len,axiom,
    ! [L: list_bool,Li2: list_VEBT_VEBT,A: fun_bo5394993412183903963T_assn] :
      ( ( size_size_list_bool(L) != size_s6755466524823107622T_VEBT(Li2) )
     => ( vEBT_L2136499394512906784T_VEBT(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_5347_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_VE1571388752190420990T_assn,A6: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5348_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_VE6910866152645720856l_assn,A6: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi) = vEBT_L4281036506115550016T_real(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5349_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_VE1323945509527881313l_assn,A6: fun_VE1323945509527881313l_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_bool,Xsi2: list_bool] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                        <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
             => ( vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi) = vEBT_L6176004506706552451T_bool(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5350_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_VE4931415397359990460t_assn,A6: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5351_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_re8050108611700907676T_assn,A6: fun_re8050108611700907676T_assn,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                   => ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi) = vEBT_L3095048238742455910T_VEBT(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5352_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_re6303261851654292474l_assn,A6: fun_re6303261851654292474l_assn,Xs: list_real,Xs4: list_real,Xsi: list_real,Xsi2: list_real] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                   => ( ( size_size_list_real(Xs) = size_size_list_real(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                        & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi) = vEBT_L5184575500739366650l_real(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5353_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_re2206215847901373695l_assn,A6: fun_re2206215847901373695l_assn,Xs: list_real,Xs4: list_real,Xsi: list_bool,Xsi2: list_bool] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                   => ( ( size_size_list_real(Xs) = size_size_list_bool(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                        & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                        <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
             => ( vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi) = vEBT_L3606510757799916617l_bool(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5354_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_re574865750389657758t_assn,A6: fun_re574865750389657758t_assn,Xs: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                   => ( ( size_size_list_real(Xs) = size_size_list_nat(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                        & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi) = vEBT_L234762979517870878al_nat(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5355_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_re560840401387337987i_assn,A6: fun_re560840401387337987i_assn,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                   => ( ( size_size_list_real(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                        & ( nth_VEBT_VEBTi(Xsi,I2) = nth_VEBT_VEBTi(Xsi2,I2) ) ) ) ) )
             => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I7,A6,Xs4),Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5356_listI__assn__weak__cong,axiom,
    ! [I6: set_nat,I7: set_nat,A: fun_bo5394993412183903963T_assn,A6: fun_bo5394993412183903963T_assn,Xs: list_bool,Xs4: list_bool,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I6 = I7 )
     => ( ( A = A6 )
       => ( ( size_size_list_bool(Xs) = size_size_list_bool(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I6))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
                   => ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( pp(aa_nat_bool(nth_bool(Xs),I2))
                        <=> pp(aa_nat_bool(nth_bool(Xs4),I2)) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L2992182275864018813T_VEBT(I6,A,Xs,Xsi) = vEBT_L2992182275864018813T_VEBT(I7,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5357_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,A6: fun_VE1571388752190420990T_assn] :
      ( ( I6 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) )
                      & ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2)) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A6,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5358_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A: fun_VE6910866152645720856l_assn,A6: fun_VE6910866152645720856l_assn] :
      ( ( I6 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) )
                      & ( aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_real(nth_real(Xsi),I2)) = aa_real_assn(aa_VEB8575627266564790985l_assn(A6,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_real(nth_real(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi) = vEBT_L4281036506115550016T_real(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5359_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_bool,Xsi2: list_bool,A: fun_VE1323945509527881313l_assn,A6: fun_VE1323945509527881313l_assn] :
      ( ( I6 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                      <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) )
                      & ( aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_bool(nth_bool(Xsi),I2)) = aa_bool_assn(aa_VEB2879522196611654106l_assn(A6,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi) = vEBT_L6176004506706552451T_bool(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5360_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A: fun_VE4931415397359990460t_assn,A6: fun_VE4931415397359990460t_assn] :
      ( ( I6 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) )
                      & ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_nat(nth_nat(Xsi),I2)) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A6,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_nat(nth_nat(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5361_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_re8050108611700907676T_assn,A6: fun_re8050108611700907676T_assn] :
      ( ( I6 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                 => ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) )
                      & ( aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2)) = aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A6,aa_nat_real(nth_real(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi) = vEBT_L3095048238742455910T_VEBT(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5362_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_real,Xsi2: list_real,A: fun_re6303261851654292474l_assn,A6: fun_re6303261851654292474l_assn] :
      ( ( I6 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                 => ( ( size_size_list_real(Xs) = size_size_list_real(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                      & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) )
                      & ( aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I2)),aa_nat_real(nth_real(Xsi),I2)) = aa_real_assn(aa_rea5239198976744917947l_assn(A6,aa_nat_real(nth_real(Xs4),I2)),aa_nat_real(nth_real(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi) = vEBT_L5184575500739366650l_real(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5363_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_bool,Xsi2: list_bool,A: fun_re2206215847901373695l_assn,A6: fun_re2206215847901373695l_assn] :
      ( ( I6 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                 => ( ( size_size_list_real(Xs) = size_size_list_bool(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                      & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                      <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) )
                      & ( aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I2)),aa_nat_bool(nth_bool(Xsi),I2)) = aa_bool_assn(aa_rea316150315452044392l_assn(A6,aa_nat_real(nth_real(Xs4),I2)),aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi) = vEBT_L3606510757799916617l_bool(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5364_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat,A: fun_re574865750389657758t_assn,A6: fun_re574865750389657758t_assn] :
      ( ( I6 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                 => ( ( size_size_list_real(Xs) = size_size_list_nat(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                      & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) )
                      & ( aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I2)),aa_nat_nat(nth_nat(Xsi),I2)) = aa_nat_assn(aa_real_fun_nat_assn(A6,aa_nat_real(nth_real(Xs4),I2)),aa_nat_nat(nth_nat(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi) = vEBT_L234762979517870878al_nat(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5365_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A: fun_re560840401387337987i_assn,A6: fun_re560840401387337987i_assn] :
      ( ( I6 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
                 => ( ( size_size_list_real(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Xs4),I2) )
                      & ( nth_VEBT_VEBTi(Xsi,I2) = nth_VEBT_VEBTi(Xsi2,I2) )
                      & ( aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I2)),nth_VEBT_VEBTi(Xsi,I2)) = aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A6,aa_nat_real(nth_real(Xs4),I2)),nth_VEBT_VEBTi(Xsi2,I2)) ) ) ) ) )
           => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I7,A6,Xs4),Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5366_listI__assn__cong,axiom,
    ! [I6: set_nat,I7: set_nat,Xs: list_bool,Xs4: list_bool,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_bo5394993412183903963T_assn,A6: fun_bo5394993412183903963T_assn] :
      ( ( I6 = I7 )
     => ( ( size_size_list_bool(Xs) = size_size_list_bool(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
                 => ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( pp(aa_nat_bool(nth_bool(Xs),I2))
                      <=> pp(aa_nat_bool(nth_bool(Xs4),I2)) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) )
                      & ( aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2)) = aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A6,aa_nat_bool(nth_bool(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2992182275864018813T_VEBT(I6,A,Xs,Xsi) = vEBT_L2992182275864018813T_VEBT(I7,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5367_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5368_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE4931415397359990460t_assn,X1: vEBT_VEBT,Xsi: list_nat,X22: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_update_nat(Xsi,I,X22)) = vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5369_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1323945509527881313l_assn,X1: vEBT_VEBT,Xsi: list_bool,X22: bool] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L6176004506706552451T_bool(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_update_bool(Xsi,I,X22)) = vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5370_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE6910866152645720856l_assn,X1: vEBT_VEBT,Xsi: list_real,X22: real] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L4281036506115550016T_real(I6,A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_update_real(Xsi,I,X22)) = vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5371_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,list_update_real(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_5372_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(I6,A,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5373_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re574865750389657758t_assn,X1: real,Xsi: list_nat,X22: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L234762979517870878al_nat(I6,A,list_update_real(Xs,I,X1),list_update_nat(Xsi,I,X22)) = vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5374_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re2206215847901373695l_assn,X1: real,Xsi: list_bool,X22: bool] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3606510757799916617l_bool(I6,A,list_update_real(Xs,I,X1),list_update_bool(Xsi,I,X22)) = vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5375_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re6303261851654292474l_assn,X1: real,Xsi: list_real,X22: real] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L5184575500739366650l_real(I6,A,list_update_real(Xs,I,X1),list_update_real(Xsi,I,X22)) = vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5376_subst__not__in,axiom,
    ! [I: nat,I6: set_nat,Xs: list_bool,A: fun_bo7535889243153246212i_assn,X1: bool,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(I6,A,list_update_bool(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(I6,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_5377_listI__assn__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi] :
      ( ( N = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(set_or4665077453230672383an_nat(zero_zero_nat,N),A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_5378_list__assn__conv__idx,axiom,
    ! [A: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(set_or4665077453230672383an_nat(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs)),A,Xs),Xsi) ).

% list_assn_conv_idx
tff(fact_5379_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5380_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5381_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1323945509527881313l_assn,Xsi: list_bool] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L6176004506706552451T_bool(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5382_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE6910866152645720856l_assn,Xsi: list_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L4281036506115550016T_real(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5383_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5384_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(insert_nat(I,I6),A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5385_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L234762979517870878al_nat(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5386_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re2206215847901373695l_assn,Xsi: list_bool] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3606510757799916617l_bool(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_bool(nth_bool(Xsi),I))),vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5387_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re6303261851654292474l_assn,Xsi: list_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L5184575500739366650l_real(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xsi),I))),vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5388_listI__assn__insert,axiom,
    ! [I: nat,I6: set_nat,Xs: list_bool,A: fun_bo5394993412183903963T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( vEBT_L2992182275864018813T_VEBT(insert_nat(I,I6),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A,aa_nat_bool(nth_bool(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2992182275864018813T_VEBT(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5389_listI__assn__conv_H,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi,F3: assn] :
      ( ( N = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(set_or4665077453230672383an_nat(zero_zero_nat,N),A,Xs),Xsi)),F3) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A,Xs),Xsi)),F3) ) ) ).

% listI_assn_conv'
tff(fact_5390_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(insert_nat(I,I6),A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,X1),X22)),vEBT_L3204528365124325536T_VEBT(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5391_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE4931415397359990460t_assn,X1: vEBT_VEBT,Xsi: list_nat,X22: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(insert_nat(I,I6),A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_update_nat(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,X1),X22)),vEBT_L8650695023172932196BT_nat(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5392_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1323945509527881313l_assn,X1: vEBT_VEBT,Xsi: list_bool,X22: bool] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L6176004506706552451T_bool(insert_nat(I,I6),A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_update_bool(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_bool_assn(aa_VEB2879522196611654106l_assn(A,X1),X22)),vEBT_L6176004506706552451T_bool(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5393_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_VEBT_VEBT,A: fun_VE6910866152645720856l_assn,X1: vEBT_VEBT,Xsi: list_real,X22: real] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L4281036506115550016T_real(insert_nat(I,I6),A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_update_real(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_real_assn(aa_VEB8575627266564790985l_assn(A,X1),X22)),vEBT_L4281036506115550016T_real(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5394_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(insert_nat(I,I6),A,list_update_real(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,X1),X22)),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I6,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5395_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(insert_nat(I,I6),A,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,X1),X22)),vEBT_L3095048238742455910T_VEBT(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5396_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re574865750389657758t_assn,X1: real,Xsi: list_nat,X22: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L234762979517870878al_nat(insert_nat(I,I6),A,list_update_real(Xs,I,X1),list_update_nat(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,X1),X22)),vEBT_L234762979517870878al_nat(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5397_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re2206215847901373695l_assn,X1: real,Xsi: list_bool,X22: bool] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3606510757799916617l_bool(insert_nat(I,I6),A,list_update_real(Xs,I,X1),list_update_bool(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_bool_assn(aa_rea316150315452044392l_assn(A,X1),X22)),vEBT_L3606510757799916617l_bool(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5398_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_real,A: fun_re6303261851654292474l_assn,X1: real,Xsi: list_real,X22: real] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L5184575500739366650l_real(insert_nat(I,I6),A,list_update_real(Xs,I,X1),list_update_real(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_real_assn(aa_rea5239198976744917947l_assn(A,X1),X22)),vEBT_L5184575500739366650l_real(I6,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5399_listI__assn__subst,axiom,
    ! [I: nat,I6: set_nat,Xs: list_bool,A: fun_bo7535889243153246212i_assn,X1: bool,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(insert_nat(I,I6),A,list_update_bool(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A,X1),X22)),aa_lis2508312872377092907i_assn(vEBT_L6372884366951783724_VEBTi(I6,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5400_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K6: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_5401_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K6: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H))),K6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H)),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),N))),H)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_5402_foldr__zero,axiom,
    ! [Xs: list_nat,D2: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D2)),D2))) ) ).

% foldr_zero
tff(fact_5403_foldr__same,axiom,
    ! [Xs: list_real,Y3: real] :
      ( ! [X4: real,Y4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => ( pp(aa_set_real_bool(member_real(Y4),set_real2(Xs)))
           => ( X4 = Y4 ) ) )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
           => ( X4 = Y3 ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y3) ) ) ) ).

% foldr_same
tff(fact_5404_lowi__hT,axiom,
    ! [X2: nat,N: nat] : pp(aa_nat_bool(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(X2,N),aa_nat_fun_nat_assn(aTP_Lamm_dh(X2),N)),one_one_nat)) ).

% lowi_hT
tff(fact_5405_highi__hT,axiom,
    ! [X2: nat,N: nat] : pp(aa_nat_bool(time_htt_nat(one_one_assn,vEBT_VEBT_highi(X2,N),aa_nat_fun_nat_assn(aTP_Lamm_di(X2),N)),one_one_nat)) ).

% highi_hT
tff(fact_5406_TBOUND__lowi,axiom,
    ! [X2: nat,N: nat] : pp(aa_nat_bool(time_TBOUND_nat(vEBT_VEBT_lowi(X2,N)),one_one_nat)) ).

% TBOUND_lowi
tff(fact_5407_TBOUND__highi,axiom,
    ! [X2: nat,N: nat] : pp(aa_nat_bool(time_TBOUND_nat(vEBT_VEBT_highi(X2,N)),one_one_nat)) ).

% TBOUND_highi
tff(fact_5408_foldr0,axiom,
    ! [Xs: list_real,C2: real,D2: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D2)),C2) ).

% foldr0
tff(fact_5409_foldr__one,axiom,
    ! [D2: nat,Ys: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D2),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D2))) ).

% foldr_one
tff(fact_5410_highi__h,axiom,
    ! [X2: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(X2,N),aa_nat_fun_nat_assn(aTP_Lamm_di(X2),N)) ).

% highi_h
tff(fact_5411_lowi__h,axiom,
    ! [X2: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(X2,N),aa_nat_fun_nat_assn(aTP_Lamm_dh(X2),N)) ).

% lowi_h
tff(fact_5412_foldr__same__int,axiom,
    ! [Xs: list_nat,Y3: nat] :
      ( ! [X4: nat,Y4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( pp(aa_set_nat_bool(member_nat(Y4),set_nat2(Xs)))
           => ( X4 = Y4 ) ) )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
           => ( X4 = Y3 ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Y3) ) ) ) ).

% foldr_same_int
tff(fact_5413_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C2: nat,D2: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Ys),I2))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C2)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D2))) ) ) ) ).

% foldr_mono
tff(fact_5414_foldr__length,axiom,
    ! [L: list_VEBT_VEBT] : aa_nat_nat(foldr_VEBT_VEBT_nat(aTP_Lamm_dj,L),zero_zero_nat) = size_s6755466524823107622T_VEBT(L) ).

% foldr_length
tff(fact_5415_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_dk,L),zero_zero_nat) = size_size_list_real(L) ).

% foldr_length
tff(fact_5416_foldr__length,axiom,
    ! [L: list_bool] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_dl,L),zero_zero_nat) = size_size_list_bool(L) ).

% foldr_length
tff(fact_5417_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_dm,L),zero_zero_nat) = size_size_list_nat(L) ).

% foldr_length
tff(fact_5418_foldr__length,axiom,
    ! [L: list_VEBT_VEBTi] : aa_nat_nat(foldr_VEBT_VEBTi_nat(aTP_Lamm_dn,L),zero_zero_nat) = size_s7982070591426661849_VEBTi(L) ).

% foldr_length
tff(fact_5419_foldr__cong,axiom,
    ! [A2: real,B2: real,L: list_real,K: list_real,F: fun_re7681423839302404990l_real,G: fun_re7681423839302404990l_real] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A3: real,X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),set_real2(L)))
             => ( aa_real_real(aa_rea7340605660300375871l_real(F,X4),A3) = aa_real_real(aa_rea7340605660300375871l_real(G,X4),A3) ) )
         => ( aa_real_real(foldr_real_real(F,L),A2) = aa_real_real(foldr_real_real(G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_5420_foldr__cong,axiom,
    ! [A2: nat,B2: nat,L: list_nat,K: list_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A3: nat,X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(L)))
             => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X4),A3) = aa_nat_nat(aa_nat_fun_nat_nat(G,X4),A3) ) )
         => ( aa_nat_nat(foldr_nat_nat(F,L),A2) = aa_nat_nat(foldr_nat_nat(G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_5421_foldr__length__aux,axiom,
    ! [L: list_VEBT_VEBT,A2: nat] : aa_nat_nat(foldr_VEBT_VEBT_nat(aTP_Lamm_dj,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_s6755466524823107622T_VEBT(L)) ).

% foldr_length_aux
tff(fact_5422_foldr__length__aux,axiom,
    ! [L: list_real,A2: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_dk,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_real(L)) ).

% foldr_length_aux
tff(fact_5423_foldr__length__aux,axiom,
    ! [L: list_bool,A2: nat] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_dl,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_bool(L)) ).

% foldr_length_aux
tff(fact_5424_foldr__length__aux,axiom,
    ! [L: list_nat,A2: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_dm,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_nat(L)) ).

% foldr_length_aux
tff(fact_5425_foldr__length__aux,axiom,
    ! [L: list_VEBT_VEBTi,A2: nat] : aa_nat_nat(foldr_VEBT_VEBTi_nat(aTP_Lamm_dn,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_s7982070591426661849_VEBTi(L)) ).

% foldr_length_aux
tff(fact_5426_norm__divide__numeral,axiom,
    ! [A2: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_5427_norm__divide__numeral,axiom,
    ! [A2: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_5428_norm__mult__numeral1,axiom,
    ! [W: num,A2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A2)) ).

% norm_mult_numeral1
tff(fact_5429_norm__mult__numeral1,axiom,
    ! [W: num,A2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A2)) ).

% norm_mult_numeral1
tff(fact_5430_norm__mult__numeral2,axiom,
    ! [A2: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_5431_norm__mult__numeral2,axiom,
    ! [A2: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_5432_norm__le__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X2)),zero_zero_real))
    <=> ( X2 = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5433_norm__le__zero__iff,axiom,
    ! [X2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X2)),zero_zero_real))
    <=> ( X2 = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5434_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_5435_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_5436_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_5437_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_5438_norm__minus__commute,axiom,
    ! [A2: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) ).

% norm_minus_commute
tff(fact_5439_norm__minus__commute,axiom,
    ! [A2: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B2),A2)) ).

% norm_minus_commute
tff(fact_5440_norm__mult,axiom,
    ! [X2: real,Y3: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y3)) ).

% norm_mult
tff(fact_5441_norm__mult,axiom,
    ! [X2: complex,Y3: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y3)) ).

% norm_mult
tff(fact_5442_norm__ge__zero,axiom,
    ! [X2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X2))) ).

% norm_ge_zero
tff(fact_5443_norm__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)) ).

% norm_divide
tff(fact_5444_norm__divide,axiom,
    ! [A2: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)) ).

% norm_divide
tff(fact_5445_norm__power,axiom,
    ! [X2: real,N: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X2)),N) ).

% norm_power
tff(fact_5446_norm__power,axiom,
    ! [X2: complex,N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X2)),N) ).

% norm_power
tff(fact_5447_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5448_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),N) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5449_nonzero__norm__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_5450_nonzero__norm__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_5451_norm__mult__less,axiom,
    ! [X2: real,R2: real,Y3: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y3)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2))) ) ) ).

% norm_mult_less
tff(fact_5452_norm__mult__less,axiom,
    ! [X2: complex,R2: real,Y3: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y3)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2))) ) ) ).

% norm_mult_less
tff(fact_5453_norm__mult__ineq,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y3)))) ).

% norm_mult_ineq
tff(fact_5454_norm__mult__ineq,axiom,
    ! [X2: complex,Y3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y3)))) ).

% norm_mult_ineq
tff(fact_5455_norm__add__less,axiom,
    ! [X2: real,R2: real,Y3: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y3)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_add_less
tff(fact_5456_norm__add__less,axiom,
    ! [X2: complex,R2: real,Y3: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y3)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_add_less
tff(fact_5457_norm__triangle__lt,axiom,
    ! [X2: real,Y3: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y3))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))),E)) ) ).

% norm_triangle_lt
tff(fact_5458_norm__triangle__lt,axiom,
    ! [X2: complex,Y3: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y3))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),Y3))),E)) ) ).

% norm_triangle_lt
tff(fact_5459_norm__triangle__le,axiom,
    ! [X2: real,Y3: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y3))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))),E)) ) ).

% norm_triangle_le
tff(fact_5460_norm__triangle__le,axiom,
    ! [X2: complex,Y3: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y3))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),Y3))),E)) ) ).

% norm_triangle_le
tff(fact_5461_norm__triangle__mono,axiom,
    ! [A2: real,R2: real,B2: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_triangle_mono
tff(fact_5462_norm__triangle__mono,axiom,
    ! [A2: complex,R2: real,B2: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A2)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),S2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2))) ) ) ).

% norm_triangle_mono
tff(fact_5463_norm__triangle__ineq,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y3)))) ).

% norm_triangle_ineq
tff(fact_5464_norm__triangle__ineq,axiom,
    ! [X2: complex,Y3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y3)))) ).

% norm_triangle_ineq
tff(fact_5465_norm__add__leD,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A2)),C2))) ) ).

% norm_add_leD
tff(fact_5466_norm__add__leD,axiom,
    ! [A2: complex,B2: complex,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2))),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),C2))) ) ).

% norm_add_leD
tff(fact_5467_norm__diff__triangle__less,axiom,
    ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5468_norm__diff__triangle__less,axiom,
    ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y3),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5469_norm__power__ineq,axiom,
    ! [X2: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X2)),N))) ).

% norm_power_ineq
tff(fact_5470_norm__power__ineq,axiom,
    ! [X2: complex,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X2)),N))) ).

% norm_power_ineq
tff(fact_5471_norm__triangle__le__diff,axiom,
    ! [X2: real,Y3: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y3))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3))),E)) ) ).

% norm_triangle_le_diff
tff(fact_5472_norm__triangle__le__diff,axiom,
    ! [X2: complex,Y3: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y3))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3))),E)) ) ).

% norm_triangle_le_diff
tff(fact_5473_norm__diff__triangle__le,axiom,
    ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_5474_norm__diff__triangle__le,axiom,
    ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y3),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_5475_norm__triangle__ineq4,axiom,
    ! [A2: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)))) ).

% norm_triangle_ineq4
tff(fact_5476_norm__triangle__ineq4,axiom,
    ! [A2: complex,B2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)))) ).

% norm_triangle_ineq4
tff(fact_5477_norm__triangle__sub,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y3)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3))))) ).

% norm_triangle_sub
tff(fact_5478_norm__triangle__sub,axiom,
    ! [X2: complex,Y3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y3)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3))))) ).

% norm_triangle_sub
tff(fact_5479_norm__diff__ineq,axiom,
    ! [A2: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)))) ).

% norm_diff_ineq
tff(fact_5480_norm__diff__ineq,axiom,
    ! [A2: complex,B2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)))) ).

% norm_diff_ineq
tff(fact_5481_norm__triangle__ineq2,axiom,
    ! [A2: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)))) ).

% norm_triangle_ineq2
tff(fact_5482_norm__triangle__ineq2,axiom,
    ! [A2: complex,B2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2)))) ).

% norm_triangle_ineq2
tff(fact_5483_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5484_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5485_norm__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D2)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2))))) ).

% norm_diff_triangle_ineq
tff(fact_5486_norm__diff__triangle__ineq,axiom,
    ! [A2: complex,B2: complex,C2: complex,D2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,C2),D2)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),C2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B2),D2))))) ).

% norm_diff_triangle_ineq
tff(fact_5487_square__norm__one,axiom,
    ! [X2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X2) = one_one_real ) ) ).

% square_norm_one
tff(fact_5488_square__norm__one,axiom,
    ! [X2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X2) = one_one_real ) ) ).

% square_norm_one
tff(fact_5489_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W))))) ) ) ).

% norm_power_diff
tff(fact_5490_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W))))) ) ) ).

% norm_power_diff
tff(fact_5491_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X4)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5492_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F: fun_real_real,Bound: real,I: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5493_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_bool,F: fun_bool_real,Bound: real,I: real] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,X4)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_bool_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_bool(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5494_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Bound: real,I: real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_nat(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5495_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_real,Bound: real,I: real] :
      ( ! [X4: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBTi_real(F,X4)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBTi_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s7982070591426661849_VEBTi(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5496_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,C2: real,G: fun_VEBT_VEBT_real,D2: real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_VEBT_VEBT_real(G,X4)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D2))) ) ).

% f_g_map_foldr_bound
tff(fact_5497_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,C2: real,G: fun_real_real,D2: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(G,X4)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D2))) ) ).

% f_g_map_foldr_bound
tff(fact_5498_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,C2: real,G: fun_nat_real,D2: real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_nat_real(G,X4)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D2))) ) ).

% f_g_map_foldr_bound
tff(fact_5499_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X4)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5500_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5501_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_bool,F: fun_bool_nat,Bound: nat,I: nat] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,X4)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_bool_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5502_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5503_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_nat,Bound: nat,I: nat] :
      ( ! [X4: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(Xs))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBTi_nat(F,X4)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBTi_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5504_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : refine4616567020171210338s_bool(vEBT_vebt_memberi(Ti,X2),vEBT_V854960066525838166emberi(T,Ti,X2)) ).

% vebt_memberi_refines
tff(fact_5505_refines__replicate,axiom,
    ! [F: heap_Time_Heap_bool,F5: heap_Time_Heap_bool,N: nat] :
      ( refine4616567020171210338s_bool(F,F5)
     => refine3902108050975373032t_bool(vEBT_V6044891547498605301i_bool(N,F),vEBT_V6044891547498605301i_bool(N,F5)) ) ).

% refines_replicate
tff(fact_5506_refines__replicate,axiom,
    ! [F: heap_T2636463487746394924on_nat,F5: heap_T2636463487746394924on_nat,N: nat] :
      ( refine7594492741263601813on_nat(F,F5)
     => refine1935026298455697829on_nat(vEBT_V792416675989592002on_nat(N,F),vEBT_V792416675989592002on_nat(N,F5)) ) ).

% refines_replicate
tff(fact_5507_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F5: heap_T8145700208782473153_VEBTi,N: nat] :
      ( refine5565527176597971370_VEBTi(F,F5)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(N,F),vEBT_V1859673955506687831_VEBTi(N,F5)) ) ).

% refines_replicate
tff(fact_5508_listsum__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,Y3: real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),Y3))) ) ).

% listsum_bound
tff(fact_5509_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Y3: real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),Y3))) ) ).

% listsum_bound
tff(fact_5510_listsum__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,Y3: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),Y3))) ) ).

% listsum_bound
tff(fact_5511_listsum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Y3: real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),Y3))) ) ).

% listsum_bound
tff(fact_5512_length__map,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(map_VE8901447254227204932T_VEBT(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5513_length__map,axiom,
    ! [F: fun_real_VEBT_VEBT,Xs: list_real] : size_s6755466524823107622T_VEBT(map_real_VEBT_VEBT(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5514_length__map,axiom,
    ! [F: fun_bool_VEBT_VEBT,Xs: list_bool] : size_s6755466524823107622T_VEBT(map_bool_VEBT_VEBT(F,Xs)) = size_size_list_bool(Xs) ).

% length_map
tff(fact_5515_length__map,axiom,
    ! [F: fun_nat_VEBT_VEBT,Xs: list_nat] : size_s6755466524823107622T_VEBT(map_nat_VEBT_VEBT(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5516_length__map,axiom,
    ! [F: fun_VE9120655849212850072T_VEBT,Xs: list_VEBT_VEBTi] : size_s6755466524823107622T_VEBT(map_VE7998069337340375161T_VEBT(F,Xs)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_map
tff(fact_5517_length__map,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : size_size_list_real(map_VEBT_VEBT_real(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5518_length__map,axiom,
    ! [F: fun_real_real,Xs: list_real] : size_size_list_real(map_real_real(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5519_length__map,axiom,
    ! [F: fun_bool_real,Xs: list_bool] : size_size_list_real(map_bool_real(F,Xs)) = size_size_list_bool(Xs) ).

% length_map
tff(fact_5520_length__map,axiom,
    ! [F: fun_nat_real,Xs: list_nat] : size_size_list_real(map_nat_real(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5521_length__map,axiom,
    ! [F: fun_VEBT_VEBTi_real,Xs: list_VEBT_VEBTi] : size_size_list_real(map_VEBT_VEBTi_real(F,Xs)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_map
tff(fact_5522_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) )
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs)))
         => ( aa_VEBT_VEBT_nat(F,X3) = aa_VEBT_VEBT_nat(G,X3) ) ) ) ).

% map_eq_conv
tff(fact_5523_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) )
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs)))
         => ( aa_VEBT_VEBT_real(F,X3) = aa_VEBT_VEBT_real(G,X3) ) ) ) ).

% map_eq_conv
tff(fact_5524_real__nat__list,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),C2)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_do(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C2)) ).

% real_nat_list
tff(fact_5525_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F,Xs)),N) = aa_VEB5832953974749077948T_VEBT(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_5526_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VE712148701811721852_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),N) = aa_VEB3973023575446460973_VEBTi(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_5527_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(nth_bool(map_VEBT_VEBT_bool(F,Xs)),N))
      <=> pp(aa_VEBT_VEBT_bool(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N))) ) ) ).

% nth_map
tff(fact_5528_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F,Xs)),N) = aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_5529_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(F,Xs)),N) = aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_5530_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_real_VEBT_VEBT(F,Xs)),N) = aa_real_VEBT_VEBT(F,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_5531_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( nth_VEBT_VEBTi(map_real_VEBT_VEBTi(F,Xs),N) = aa_real_VEBT_VEBTi(F,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_5532_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( aa_nat_nat(nth_nat(map_real_nat(F,Xs)),N) = aa_real_nat(F,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_5533_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( pp(aa_nat_bool(nth_bool(map_real_bool(F,Xs)),N))
      <=> pp(aa_real_bool(F,aa_nat_real(nth_real(Xs),N))) ) ) ).

% nth_map
tff(fact_5534_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( aa_nat_real(nth_real(map_real_real(F,Xs)),N) = aa_real_real(F,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_5535_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: fun_bo7602104967519523767p_bool,F12: fun_bo7602104967519523767p_bool,F22: fun_op3392518339485081117p_bool,F23: fun_op3392518339485081117p_bool] :
      ( ( Ti = Ti2 )
     => ( ! [A3: bool,B3: bool] : refine4616567020171210338s_bool(aa_boo5261840290332707324p_bool(aa_boo5285575089431429040p_bool(F1,A3),B3),aa_boo5261840290332707324p_bool(aa_boo5285575089431429040p_bool(F12,A3),B3))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : refine4616567020171210338s_bool(aa_VEB101718416255554708p_bool(aa_arr628998674462452489p_bool(aa_nat5560262748439664842p_bool(aa_opt8498226628657232662p_bool(F22,Info2),Deg2),TreeArray),Summary2),aa_VEB101718416255554708p_bool(aa_arr628998674462452489p_bool(aa_nat5560262748439664842p_bool(aa_opt8498226628657232662p_bool(F23,Info2),Deg2),TreeArray),Summary2))
         => refine4616567020171210338s_bool(vEBT_c4579152981172362421p_bool(F22,F1,Ti),vEBT_c4579152981172362421p_bool(F23,F12,Ti2)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5536_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: fun_bo8430231047727935170on_nat,F12: fun_bo8430231047727935170on_nat,F22: fun_op5783775701855621896on_nat,F23: fun_op5783775701855621896on_nat] :
      ( ( Ti = Ti2 )
     => ( ! [A3: bool,B3: bool] : refine7594492741263601813on_nat(aa_boo4584856347283357585on_nat(aa_boo3438790501838819827on_nat(F1,A3),B3),aa_boo4584856347283357585on_nat(aa_boo3438790501838819827on_nat(F12,A3),B3))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F22,Info2),Deg2),TreeArray),Summary2),aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F23,Info2),Deg2),TreeArray),Summary2))
         => refine7594492741263601813on_nat(vEBT_c6250501799366334488on_nat(F22,F1,Ti),vEBT_c6250501799366334488on_nat(F23,F12,Ti2)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5537_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: fun_bo851793448214054487_VEBTi,F12: fun_bo851793448214054487_VEBTi,F22: fun_op5127434927081926045_VEBTi,F23: fun_op5127434927081926045_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A3: bool,B3: bool] : refine5565527176597971370_VEBTi(aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F1,A3),B3),aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F12,A3),B3))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info2),Deg2),TreeArray),Summary2),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F23,Info2),Deg2),TreeArray),Summary2))
         => refine5565527176597971370_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F1,Ti),vEBT_c6028912655521741485_VEBTi(F23,F12,Ti2)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5538_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5539_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5540_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5541_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5542_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_bool_nat,Ys: list_bool] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_bool_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5543_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_bool_real,Ys: list_bool] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_bool_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5544_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5545_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5546_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBTi_nat,Ys: list_VEBT_VEBTi] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBTi_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s7982070591426661849_VEBTi(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5547_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBTi_real,Ys: list_VEBT_VEBTi] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBTi_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s7982070591426661849_VEBTi(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5548_list_Omap__cong,axiom,
    ! [X2: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( X2 = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Ya)))
           => ( aa_VEBT_VEBT_nat(F,Z2) = aa_VEBT_VEBT_nat(G,Z2) ) )
       => ( map_VEBT_VEBT_nat(F,X2) = map_VEBT_VEBT_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5549_list_Omap__cong,axiom,
    ! [X2: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( X2 = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Ya)))
           => ( aa_VEBT_VEBT_real(F,Z2) = aa_VEBT_VEBT_real(G,Z2) ) )
       => ( map_VEBT_VEBT_real(F,X2) = map_VEBT_VEBT_real(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5550_list_Omap__cong0,axiom,
    ! [X2: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [Z2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(X2)))
         => ( aa_VEBT_VEBT_nat(F,Z2) = aa_VEBT_VEBT_nat(G,Z2) ) )
     => ( map_VEBT_VEBT_nat(F,X2) = map_VEBT_VEBT_nat(G,X2) ) ) ).

% list.map_cong0
tff(fact_5551_list_Omap__cong0,axiom,
    ! [X2: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [Z2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(X2)))
         => ( aa_VEBT_VEBT_real(F,Z2) = aa_VEBT_VEBT_real(G,Z2) ) )
     => ( map_VEBT_VEBT_real(F,X2) = map_VEBT_VEBT_real(G,X2) ) ) ).

% list.map_cong0
tff(fact_5552_list_Oinj__map__strong,axiom,
    ! [X2: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Fa: fun_VEBT_VEBT_nat] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(X2)))
         => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xa)))
           => ( ( aa_VEBT_VEBT_nat(F,Z2) = aa_VEBT_VEBT_nat(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_nat(F,X2) = map_VEBT_VEBT_nat(Fa,Xa) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
tff(fact_5553_list_Oinj__map__strong,axiom,
    ! [X2: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Fa: fun_VEBT_VEBT_real] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(X2)))
         => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xa)))
           => ( ( aa_VEBT_VEBT_real(F,Z2) = aa_VEBT_VEBT_real(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_real(F,X2) = map_VEBT_VEBT_real(Fa,Xa) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
tff(fact_5554_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => ( aa_VEBT_VEBT_nat(F,X4) = aa_VEBT_VEBT_nat(G,X4) ) )
     => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) ) ) ).

% map_ext
tff(fact_5555_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => ( aa_VEBT_VEBT_real(F,X4) = aa_VEBT_VEBT_real(G,X4) ) )
     => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) ) ) ).

% map_ext
tff(fact_5556_map__idI,axiom,
    ! [Xs: list_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => ( aa_int_int(F,X4) = X4 ) )
     => ( map_int_int(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5557_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => ( aa_VEB5832953974749077948T_VEBT(F,X4) = X4 ) )
     => ( map_VE8901447254227204932T_VEBT(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5558_map__idI,axiom,
    ! [Xs: list_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => ( aa_real_real(F,X4) = X4 ) )
     => ( map_real_real(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5559_map__idI,axiom,
    ! [Xs: list_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( aa_nat_nat(F,X4) = X4 ) )
     => ( map_nat_nat(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5560_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Ys)))
           => ( aa_VEBT_VEBT_nat(F,X4) = aa_VEBT_VEBT_nat(G,X4) ) )
       => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_5561_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Ys)))
           => ( aa_VEBT_VEBT_real(F,X4) = aa_VEBT_VEBT_real(G,X4) ) )
       => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) ) ) ) ).

% map_cong
tff(fact_5562_ex__map__conv,axiom,
    ! [Ys: list_real,F: fun_VEBT_VEBT_real] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_real(F,Xs3)
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Ys)))
         => ? [Xa4: vEBT_VEBT] : X3 = aa_VEBT_VEBT_real(F,Xa4) ) ) ).

% ex_map_conv
tff(fact_5563_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_VEBT_VEBT_nat] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_nat(F,Xs3)
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Ys)))
         => ? [Xa4: vEBT_VEBT] : X3 = aa_VEBT_VEBT_nat(F,Xa4) ) ) ).

% ex_map_conv
tff(fact_5564_map__eq__nth__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,L: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_nat(F,L) = map_VEBT_VEBT_nat(F,L3) )
     => ( aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_5565_map__eq__nth__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,L: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_real(F,L) = map_VEBT_VEBT_real(F,L3) )
     => ( aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_5566_map__update,axiom,
    ! [F: fun_VE8449866691444886503_VEBTi,Xs: list_VEBT_VEBTi,K: nat,Y3: vEBT_VEBTi] : map_VE483055756984248624_VEBTi(F,list_u6098035379799741383_VEBTi(Xs,K,Y3)) = list_u6098035379799741383_VEBTi(map_VE483055756984248624_VEBTi(F,Xs),K,aa_VEB5586453056816421800_VEBTi(F,Y3)) ).

% map_update
tff(fact_5567_map__update,axiom,
    ! [F: fun_VE9120655849212850072T_VEBT,Xs: list_VEBT_VEBTi,K: nat,Y3: vEBT_VEBTi] : map_VE7998069337340375161T_VEBT(F,list_u6098035379799741383_VEBTi(Xs,K,Y3)) = list_u1324408373059187874T_VEBT(map_VE7998069337340375161T_VEBT(F,Xs),K,aa_VEB4941942288398148609T_VEBT(F,Y3)) ).

% map_update
tff(fact_5568_map__update,axiom,
    ! [F: fun_VEBT_VEBTi_nat,Xs: list_VEBT_VEBTi,K: nat,Y3: vEBT_VEBTi] : map_VEBT_VEBTi_nat(F,list_u6098035379799741383_VEBTi(Xs,K,Y3)) = list_update_nat(map_VEBT_VEBTi_nat(F,Xs),K,aa_VEBT_VEBTi_nat(F,Y3)) ).

% map_update
tff(fact_5569_map__update,axiom,
    ! [F: fun_VEBT_VEBTi_bool,Xs: list_VEBT_VEBTi,K: nat,Y3: vEBT_VEBTi] : map_VEBT_VEBTi_bool(F,list_u6098035379799741383_VEBTi(Xs,K,Y3)) = list_update_bool(map_VEBT_VEBTi_bool(F,Xs),K,aa_VEBT_VEBTi_bool(F,Y3)) ).

% map_update
tff(fact_5570_map__update,axiom,
    ! [F: fun_VEBT_VEBTi_real,Xs: list_VEBT_VEBTi,K: nat,Y3: vEBT_VEBTi] : map_VEBT_VEBTi_real(F,list_u6098035379799741383_VEBTi(Xs,K,Y3)) = list_update_real(map_VEBT_VEBTi_real(F,Xs),K,aa_VEBT_VEBTi_real(F,Y3)) ).

% map_update
tff(fact_5571_map__update,axiom,
    ! [F: fun_VE712148701811721852_VEBTi,Xs: list_VEBT_VEBT,K: nat,Y3: vEBT_VEBT] : map_VE7029150624388687525_VEBTi(F,list_u1324408373059187874T_VEBT(Xs,K,Y3)) = list_u6098035379799741383_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),K,aa_VEB3973023575446460973_VEBTi(F,Y3)) ).

% map_update
tff(fact_5572_map__update,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,K: nat,Y3: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,list_u1324408373059187874T_VEBT(Xs,K,Y3)) = list_u1324408373059187874T_VEBT(map_VE8901447254227204932T_VEBT(F,Xs),K,aa_VEB5832953974749077948T_VEBT(F,Y3)) ).

% map_update
tff(fact_5573_map__update,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,K: nat,Y3: vEBT_VEBT] : map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(Xs,K,Y3)) = list_update_nat(map_VEBT_VEBT_nat(F,Xs),K,aa_VEBT_VEBT_nat(F,Y3)) ).

% map_update
tff(fact_5574_map__update,axiom,
    ! [F: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT,K: nat,Y3: vEBT_VEBT] : map_VEBT_VEBT_bool(F,list_u1324408373059187874T_VEBT(Xs,K,Y3)) = list_update_bool(map_VEBT_VEBT_bool(F,Xs),K,aa_VEBT_VEBT_bool(F,Y3)) ).

% map_update
tff(fact_5575_map__update,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,K: nat,Y3: vEBT_VEBT] : map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(Xs,K,Y3)) = list_update_real(map_VEBT_VEBT_real(F,Xs),K,aa_VEBT_VEBT_real(F,Y3)) ).

% map_update
tff(fact_5576_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,X2: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_nat(F,X2) ) )
     => ( map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(L,I,X2)) = map_VEBT_VEBT_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_5577_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,X2: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_real(F,X2) ) )
     => ( map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(L,I,X2)) = map_VEBT_VEBT_real(F,L) ) ) ).

% map_upd_eq
tff(fact_5578_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_5579_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) ).

% VEBT_internal.cnt.simps(2)
tff(fact_5580_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,X2) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_5581_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,X2) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y3 != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_5582_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_5583_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,X2) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5584_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_5585_VEBT__internal_Ospace_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,X2) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5586_round__unique,axiom,
    ! [X2: real,Y3: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X2) = Y3 ) ) ) ).

% round_unique
tff(fact_5587_round__unique,axiom,
    ! [X2: rat,Y3: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y3)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X2) = Y3 ) ) ) ).

% round_unique
tff(fact_5588_mult__le__cancel__iff1,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ).

% mult_le_cancel_iff1
tff(fact_5589_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(Y3),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ) ).

% mult_le_cancel_iff1
tff(fact_5590_mult__le__cancel__iff1,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ) ).

% mult_le_cancel_iff1
tff(fact_5591_mult__le__cancel__iff2,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ).

% mult_le_cancel_iff2
tff(fact_5592_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X2)),aa_rat_rat(times_times_rat(Z),Y3)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ) ).

% mult_le_cancel_iff2
tff(fact_5593_mult__le__cancel__iff2,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y3)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ) ).

% mult_le_cancel_iff2
tff(fact_5594_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_succi(Ti,X2),vEBT_VEBT_vebt_succi(T,Ti,X2)) ).

% vebt_succi_refines
tff(fact_5595_vebt__predi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_predi(Ti,X2),vEBT_VEBT_vebt_predi(T,Ti,X2)) ).

% vebt_predi_refines
tff(fact_5596_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5597_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5598_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5599_vebt__buildupi__refines,axiom,
    ! [N: nat] : refine5565527176597971370_VEBTi(vEBT_vebt_buildupi(N),vEBT_V739175172307565963ildupi(N)) ).

% vebt_buildupi_refines
tff(fact_5600_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_inserti(Ti,X2),vEBT_V3964819847710782039nserti(T,Ti,X2)) ).

% vebt_inserti_refines
tff(fact_5601_round__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5602_round__numeral,axiom,
    ! [N: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5603_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5604_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_5605_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5606_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5607_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,M)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5608_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5609_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5610_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5611_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5612_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5613_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5614_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5615_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5616_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5617_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo851793448214054487_VEBTi,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op5127434927081926045_VEBTi,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: bool,B3: bool] :
          ( ( Ti = vEBT_Leafi(A3,B3) )
         => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F,A3),B3)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A3),B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_c6028912655521741485_VEBTi(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_5618_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo7602104967519523767p_bool,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op3392518339485081117p_bool,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: bool,B3: bool] :
          ( ( Ti = vEBT_Leafi(A3,B3) )
         => pp(aa_nat_bool(time_TBOUND_bool(aa_boo5261840290332707324p_bool(aa_boo5285575089431429040p_bool(F,A3),B3)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A3),B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_TBOUND_bool(aa_VEB101718416255554708p_bool(aa_arr628998674462452489p_bool(aa_nat5560262748439664842p_bool(aa_opt8498226628657232662p_bool(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_TBOUND_bool(vEBT_c4579152981172362421p_bool(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_5619_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo8430231047727935170on_nat,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op5783775701855621896on_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: bool,B3: bool] :
          ( ( Ti = vEBT_Leafi(A3,B3) )
         => pp(aa_nat_bool(time_T8353473612707095248on_nat(aa_boo4584856347283357585on_nat(aa_boo3438790501838819827on_nat(F,A3),B3)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A3),B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_c6250501799366334488on_nat(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_5620_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo5340758216624605938ap_nat,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op2425996437410981688ap_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: bool,B3: bool] :
          ( ( Ti = vEBT_Leafi(A3,B3) )
         => pp(aa_nat_bool(time_TBOUND_nat(aa_boo3049335359029405505ap_nat(aa_boo988204379142029347ap_nat(F,A3),B3)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A3),B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_TBOUND_nat(aa_VEB183921879236458153ap_nat(aa_arr6934346397072338570ap_nat(aa_nat1288005622339309855ap_nat(aa_opt6697879614838309225ap_nat(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_TBOUND_nat(vEBT_c1335663792808957512ap_nat(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_5621_round__mono,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X2)),aa_rat_int(archim7778729529865785530nd_rat,Y3))) ) ).

% round_mono
tff(fact_5622_ceiling__ge__round,axiom,
    ! [X2: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X2)),aa_real_int(archim7802044766580827645g_real,X2))) ).

% ceiling_ge_round
tff(fact_5623_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5624_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5625_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5626_mult__less__iff1,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ).

% mult_less_iff1
tff(fact_5627_mult__less__iff1,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(Y3),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ) ).

% mult_less_iff1
tff(fact_5628_mult__less__iff1,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y3),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y3)) ) ) ).

% mult_less_iff1
tff(fact_5629_of__int__round__le,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_5630_of__int__round__le,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X2))),aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_5631_of__int__round__ge,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X2)))) ).

% of_int_round_ge
tff(fact_5632_of__int__round__ge,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X2)))) ).

% of_int_round_ge
tff(fact_5633_of__int__round__gt,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X2)))) ).

% of_int_round_gt
tff(fact_5634_of__int__round__gt,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X2)))) ).

% of_int_round_gt
tff(fact_5635_divides__aux__eq,axiom,
    ! [Q3: code_integer,R2: code_integer] :
      ( pp(aa_Pro5843519176478241656r_bool(unique5706413561485394159nteger,produc1086072967326762835nteger(Q3,R2)))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_5636_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_5637_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_5638_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X2: nat,Y3: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != one_one_int ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != one_one_int ) )
         => ~ ! [N3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_5639_highi__def,axiom,
    ! [X2: nat,N: nat] : vEBT_VEBT_highi(X2,N) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% highi_def
tff(fact_5640_nth__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A2: array_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => hoare_1429296392585015714_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A2),Xs),array_nth_VEBT_VEBTi(A2,I),aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_dp(I),Xs),A2)) ) ).

% nth_rule
tff(fact_5641_nth__rule,axiom,
    ! [I: nat,Xs: list_bool,A2: array_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => hoare_2774773592644289642e_bool(snga_assn_bool(A2,Xs),array_nth_bool(A2,I),aa_arr3483125744708554596l_assn(aa_lis2577335236791371354l_assn(aTP_Lamm_dq(I),Xs),A2)) ) ).

% nth_rule
tff(fact_5642_nth__rule,axiom,
    ! [I: nat,Xs: list_option_nat,A2: array_option_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6086282163384603972on_nat(Xs)))
     => hoare_7629718768684598413on_nat(snga_assn_option_nat(A2,Xs),array_nth_option_nat(A2,I),aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_dr(I),Xs),A2)) ) ).

% nth_rule
tff(fact_5643_nth__rule,axiom,
    ! [I: nat,Xs: list_nat,A2: array_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => hoare_3067605981109127869le_nat(snga_assn_nat(A2,Xs),array_nth_nat(A2,I),aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ds(I),Xs),A2)) ) ).

% nth_rule
tff(fact_5644_unset__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_5645_unset__bit__0,axiom,
    ! [A2: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_5646_fi__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ps: assn,F3: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(times_times_assn(P),F3)))
       => hoare_1429296392585015714_VEBTi(Ps,C2,aa_ass4389913640333755968i_assn(aTP_Lamm_bp(Q),F3)) ) ) ).

% fi_rule
tff(fact_5647_fi__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn,Ps: assn,F3: assn] :
      ( hoare_2774773592644289642e_bool(P,C2,Q)
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(times_times_assn(P),F3)))
       => hoare_2774773592644289642e_bool(Ps,C2,aa_ass2376637134808068204l_assn(aTP_Lamm_bq(Q),F3)) ) ) ).

% fi_rule
tff(fact_5648_fi__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ps: assn,F3: assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(times_times_assn(P),F3)))
       => hoare_7629718768684598413on_nat(Ps,C2,aa_ass4112278883933122603t_assn(aTP_Lamm_br(Q),F3)) ) ) ).

% fi_rule
tff(fact_5649_fi__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,Ps: assn,F3: assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(times_times_assn(P),F3)))
       => hoare_3067605981109127869le_nat(Ps,C2,aa_assn_fun_nat_assn(aTP_Lamm_bs(Q),F3)) ) ) ).

% fi_rule
tff(fact_5650_highsimp,axiom,
    ! [X2: nat,N: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N)) = vEBT_VEBT_highi(X2,N) ).

% highsimp
tff(fact_5651_lowsimp,axiom,
    ! [X2: nat,N: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) = vEBT_VEBT_lowi(X2,N) ).

% lowsimp
tff(fact_5652_dvd__add__triv__right__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_5653_dvd__add__triv__right__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_5654_dvd__add__triv__right__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_5655_dvd__add__triv__right__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ).

% dvd_add_triv_right_iff
tff(fact_5656_dvd__add__triv__left__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_5657_dvd__add__triv__left__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(A2),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_5658_dvd__add__triv__left__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_5659_dvd__add__triv__left__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ).

% dvd_add_triv_left_iff
tff(fact_5660_div__dvd__div,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C2)) ) ) ) ).

% div_dvd_div
tff(fact_5661_div__dvd__div,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A2)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C2)) ) ) ) ).

% div_dvd_div
tff(fact_5662_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_5663_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_5664_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C2)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5665_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C2)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5666_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C2)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5667_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C2)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5668_dvd__mult__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),C2)))
    <=> ( ( C2 = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A2),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5669_dvd__mult__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)))
    <=> ( ( C2 = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5670_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)))
    <=> ( ( C2 = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5671_dvd__mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
    <=> ( ( C2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5672_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),B2)))
    <=> ( ( C2 = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A2),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5673_dvd__mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)))
    <=> ( ( C2 = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5674_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)))
    <=> ( ( C2 = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5675_dvd__mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
    <=> ( ( C2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5676_dvd__add__times__triv__right__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5677_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(times_times_rat(C2),A2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5678_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5679_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5680_dvd__add__times__triv__left__iff,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5681_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C2),A2)),B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5682_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5683_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5684_unit__prod,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_5685_unit__prod,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int)) ) ) ).

% unit_prod
tff(fact_5686_div__add,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ) ) ).

% div_add
tff(fact_5687_div__add,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ) ).

% div_add
tff(fact_5688_unit__div,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat)) ) ) ).

% unit_div
tff(fact_5689_unit__div,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int)) ) ) ).

% unit_div
tff(fact_5690_unit__div__1__unit,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)),one_one_nat)) ) ).

% unit_div_1_unit
tff(fact_5691_unit__div__1__unit,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_5692_unit__div__1__div__1,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5693_unit__div__1__div__1,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5694_dvd__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5695_dvd__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5696_dvd__mult__div__cancel,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5697_dvd__mult__div__cancel,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5698_div__diff,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ) ).

% div_diff
tff(fact_5699_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_5700_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_5701_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_5702_unit__mult__div__div,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5703_unit__mult__div__div,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5704_unit__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5705_unit__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5706_odd__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) ) ) ).

% odd_add
tff(fact_5707_odd__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) ) ) ).

% odd_add
tff(fact_5708_even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) ) ) ).

% even_add
tff(fact_5709_even__add,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) ) ) ).

% even_add
tff(fact_5710_even__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) ) ) ).

% even_mult_iff
tff(fact_5711_even__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) ) ) ).

% even_mult_iff
tff(fact_5712_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_5713_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_5714_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(N,M))) ) ).

% dvd_numeral_simp
tff(fact_5715_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,unique5052692396658037445od_int(N,M))) ) ).

% dvd_numeral_simp
tff(fact_5716_even__plus__one__iff,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)) ) ).

% even_plus_one_iff
tff(fact_5717_even__plus__one__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) ) ).

% even_plus_one_iff
tff(fact_5718_even__diff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ).

% even_diff
tff(fact_5719_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_5720_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_5721_even__succ__div__2,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5722_even__succ__div__2,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5723_odd__succ__div__two,axiom,
    ! [A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_5724_odd__succ__div__two,axiom,
    ! [A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_5725_even__succ__div__two,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5726_even__succ__div__two,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5727_even__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_5728_even__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_5729_even__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_5730_zero__le__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5731_zero__le__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5732_zero__le__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5733_zero__le__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5734_power__less__zero__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),zero_z3403309356797280102nteger))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ) ).

% power_less_zero_eq
tff(fact_5735_power__less__zero__eq,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_5736_power__less__zero__eq,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_5737_power__less__zero__eq,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_5738_power__less__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5739_power__less__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5740_power__less__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5741_power__less__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5742_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_5743_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_5744_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_5745_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).

% even_diff_nat
tff(fact_5746_odd__two__times__div__two__succ,axiom,
    ! [A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_5747_odd__two__times__div__two__succ,axiom,
    ! [A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_5748_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5749_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5750_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5751_zero__less__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5752_zero__less__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A2 != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5753_zero__less__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A2 != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5754_zero__less__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A2 != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5755_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_5756_power__le__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5757_power__le__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5758_power__le__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5759_power__le__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5760_even__succ__div__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5761_even__succ__div__exp,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5762_even__succ__div__exp,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5763_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_5764_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_5765_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_5766_dvd__add__right__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5767_dvd__add__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5768_dvd__add__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5769_dvd__add__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5770_dvd__add__left__iff,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_5771_dvd__add__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_5772_dvd__add__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_5773_dvd__add__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ) ).

% dvd_add_left_iff
tff(fact_5774_dvd__add,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))) ) ) ).

% dvd_add
tff(fact_5775_dvd__add,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),C2))) ) ) ).

% dvd_add
tff(fact_5776_dvd__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))) ) ) ).

% dvd_add
tff(fact_5777_dvd__add,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C2))) ) ) ).

% dvd_add
tff(fact_5778_dvd__unit__imp__unit,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_5779_dvd__unit__imp__unit,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_5780_unit__imp__dvd,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2)) ) ).

% unit_imp_dvd
tff(fact_5781_unit__imp__dvd,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2)) ) ).

% unit_imp_dvd
tff(fact_5782_one__dvd,axiom,
    ! [A2: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(dvd_dvd_assn,one_one_assn),A2)) ).

% one_dvd
tff(fact_5783_one__dvd,axiom,
    ! [A2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A2)) ).

% one_dvd
tff(fact_5784_one__dvd,axiom,
    ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A2)) ).

% one_dvd
tff(fact_5785_one__dvd,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A2)) ).

% one_dvd
tff(fact_5786_one__dvd,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A2)) ).

% one_dvd
tff(fact_5787_dvdE,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A2))
     => ~ ! [K2: real] : A2 != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K2) ) ).

% dvdE
tff(fact_5788_dvdE,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B2),A2))
     => ~ ! [K2: rat] : A2 != aa_rat_rat(times_times_rat(B2),K2) ) ).

% dvdE
tff(fact_5789_dvdE,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
     => ~ ! [K2: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K2) ) ).

% dvdE
tff(fact_5790_dvdE,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
     => ~ ! [K2: int] : A2 != aa_int_int(aa_int_fun_int_int(times_times_int,B2),K2) ) ).

% dvdE
tff(fact_5791_dvdI,axiom,
    ! [A2: real,B2: real,K: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A2)) ) ).

% dvdI
tff(fact_5792_dvdI,axiom,
    ! [A2: rat,B2: rat,K: rat] :
      ( ( A2 = aa_rat_rat(times_times_rat(B2),K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B2),A2)) ) ).

% dvdI
tff(fact_5793_dvdI,axiom,
    ! [A2: nat,B2: nat,K: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2)) ) ).

% dvdI
tff(fact_5794_dvdI,axiom,
    ! [A2: int,B2: int,K: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2)) ) ).

% dvdI
tff(fact_5795_dvd__def,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A2))
    <=> ? [K3: real] : A2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K3) ) ).

% dvd_def
tff(fact_5796_dvd__def,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B2),A2))
    <=> ? [K3: rat] : A2 = aa_rat_rat(times_times_rat(B2),K3) ) ).

% dvd_def
tff(fact_5797_dvd__def,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
    <=> ? [K3: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K3) ) ).

% dvd_def
tff(fact_5798_dvd__def,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
    <=> ? [K3: int] : A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K3) ) ).

% dvd_def
tff(fact_5799_dvd__mult,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ) ).

% dvd_mult
tff(fact_5800_dvd__mult,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),C2))) ) ).

% dvd_mult
tff(fact_5801_dvd__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))) ) ).

% dvd_mult
tff(fact_5802_dvd__mult,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))) ) ).

% dvd_mult
tff(fact_5803_dvd__mult2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ) ).

% dvd_mult2
tff(fact_5804_dvd__mult2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),C2))) ) ).

% dvd_mult2
tff(fact_5805_dvd__mult2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))) ) ).

% dvd_mult2
tff(fact_5806_dvd__mult2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))) ) ).

% dvd_mult2
tff(fact_5807_dvd__mult__left,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),C2)) ) ).

% dvd_mult_left
tff(fact_5808_dvd__mult__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),B2)),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),C2)) ) ).

% dvd_mult_left
tff(fact_5809_dvd__mult__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ).

% dvd_mult_left
tff(fact_5810_dvd__mult__left,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ).

% dvd_mult_left
tff(fact_5811_dvd__triv__left,axiom,
    ! [A2: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))) ).

% dvd_triv_left
tff(fact_5812_dvd__triv__left,axiom,
    ! [A2: rat,B2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(A2),B2))) ).

% dvd_triv_left
tff(fact_5813_dvd__triv__left,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))) ).

% dvd_triv_left
tff(fact_5814_dvd__triv__left,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ).

% dvd_triv_left
tff(fact_5815_mult__dvd__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),D2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))) ) ) ).

% mult_dvd_mono
tff(fact_5816_mult__dvd__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),B2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),D2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2))) ) ) ).

% mult_dvd_mono
tff(fact_5817_mult__dvd__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ).

% mult_dvd_mono
tff(fact_5818_mult__dvd__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ).

% mult_dvd_mono
tff(fact_5819_dvd__mult__right,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),C2)) ) ).

% dvd_mult_right
tff(fact_5820_dvd__mult__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),B2)),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B2),C2)) ) ).

% dvd_mult_right
tff(fact_5821_dvd__mult__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C2)) ) ).

% dvd_mult_right
tff(fact_5822_dvd__mult__right,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C2)) ) ).

% dvd_mult_right
tff(fact_5823_dvd__triv__right,axiom,
    ! [A2: real,B2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2))) ).

% dvd_triv_right
tff(fact_5824_dvd__triv__right,axiom,
    ! [A2: rat,B2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),A2))) ).

% dvd_triv_right
tff(fact_5825_dvd__triv__right,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2))) ).

% dvd_triv_right
tff(fact_5826_dvd__triv__right,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2))) ).

% dvd_triv_right
tff(fact_5827_dvd__diff__commute,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C2))) ) ).

% dvd_diff_commute
tff(fact_5828_dvd__diff,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),Z))) ) ) ).

% dvd_diff
tff(fact_5829_dvd__diff,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),Y3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),aa_rat_rat(minus_minus_rat(Y3),Z))) ) ) ).

% dvd_diff
tff(fact_5830_dvd__diff,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y3),Z))) ) ) ).

% dvd_diff
tff(fact_5831_dvd__div__eq__iff,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),A2))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),B2))
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5832_dvd__div__eq__iff,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),B2))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5833_dvd__div__eq__iff,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),B2))
       => ( ( aa_rat_rat(divide_divide_rat(A2),C2) = aa_rat_rat(divide_divide_rat(B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5834_dvd__div__eq__iff,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5835_dvd__div__eq__iff,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5836_dvd__div__eq__cancel,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C2) )
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),A2))
       => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),B2))
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5837_dvd__div__eq__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),A2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),B2))
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5838_dvd__div__eq__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),C2) = aa_rat_rat(divide_divide_rat(B2),C2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),A2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),B2))
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5839_dvd__div__eq__cancel,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5840_dvd__div__eq__cancel,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5841_div__div__div__same,axiom,
    ! [D2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_5842_div__div__div__same,axiom,
    ! [D2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),D2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_5843_dvd__power__same,axiom,
    ! [X2: nat,Y3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),Y3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),N))) ) ).

% dvd_power_same
tff(fact_5844_dvd__power__same,axiom,
    ! [X2: real,Y3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N))) ) ).

% dvd_power_same
tff(fact_5845_dvd__power__same,axiom,
    ! [X2: int,Y3: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),N))) ) ).

% dvd_power_same
tff(fact_5846_dvd__power__same,axiom,
    ! [X2: complex,Y3: complex,N: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X2),Y3))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),N))) ) ).

% dvd_power_same
tff(fact_5847_dvd__power__same,axiom,
    ! [X2: code_integer,Y3: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),Y3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),N))) ) ).

% dvd_power_same
tff(fact_5848_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_5849_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)),K)) ).

% unset_bit_less_eq
tff(fact_5850_subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dt,A2))),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dt,B2))))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A2),B2)) ) ).

% subset_divisors_dvd
tff(fact_5851_subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_du,A2))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_du,B2))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ).

% subset_divisors_dvd
tff(fact_5852_subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_dv,A2))),collect_int(aa_int_fun_int_bool(aTP_Lamm_dv,B2))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ).

% subset_divisors_dvd
tff(fact_5853_strict__subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dt,A2))),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dt,B2))))
    <=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A2),B2))
        & ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),A2)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5854_strict__subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_du,A2))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_du,B2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5855_strict__subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_dv,A2))),collect_int(aa_int_fun_int_bool(aTP_Lamm_dv,B2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5856_not__is__unit__0,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).

% not_is_unit_0
tff(fact_5857_not__is__unit__0,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).

% not_is_unit_0
tff(fact_5858_pinf_I9_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2))) ) ) ).

% pinf(9)
tff(fact_5859_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2))) ) ) ).

% pinf(9)
tff(fact_5860_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2))) ) ) ).

% pinf(9)
tff(fact_5861_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2))) ) ) ).

% pinf(9)
tff(fact_5862_pinf_I10_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X6))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2))) ) ) ).

% pinf(10)
tff(fact_5863_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X6))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2))) ) ) ).

% pinf(10)
tff(fact_5864_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X6))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2))) ) ) ).

% pinf(10)
tff(fact_5865_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X6))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2))) ) ) ).

% pinf(10)
tff(fact_5866_minf_I9_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2))) ) ) ).

% minf(9)
tff(fact_5867_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2))) ) ) ).

% minf(9)
tff(fact_5868_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2))) ) ) ).

% minf(9)
tff(fact_5869_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2))) ) ) ).

% minf(9)
tff(fact_5870_minf_I10_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),Z2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),S2))) ) ) ).

% minf(10)
tff(fact_5871_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),Z2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),S2))) ) ) ).

% minf(10)
tff(fact_5872_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X6),S2))) ) ) ).

% minf(10)
tff(fact_5873_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),S2))) ) ) ).

% minf(10)
tff(fact_5874_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B2),A2))
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) = zero_zero_complex )
      <=> ( A2 = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5875_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B2),A2))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = zero_zero_real )
      <=> ( A2 = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5876_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B2),A2))
     => ( ( aa_rat_rat(divide_divide_rat(A2),B2) = zero_zero_rat )
      <=> ( A2 = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5877_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat )
      <=> ( A2 = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5878_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5879_unit__mult__right__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_5880_unit__mult__right__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_5881_unit__mult__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_5882_unit__mult__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_5883_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5884_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5885_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5886_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5887_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_5888_mult__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_5889_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_5890_dvd__mult__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_5891_is__unit__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_5892_is__unit__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_5893_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5894_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5895_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5896_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5897_unit__div__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_div_cancel
tff(fact_5898_unit__div__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_div_cancel
tff(fact_5899_div__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_5900_div__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_5901_dvd__div__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),B2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_5902_dvd__div__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_5903_dvd__div__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),C2) ) ) ).

% dvd_div_mult
tff(fact_5904_dvd__div__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),C2) ) ) ).

% dvd_div_mult
tff(fact_5905_div__mult__swap,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ) ) ).

% div_mult_swap
tff(fact_5906_div__mult__swap,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) ) ) ).

% div_mult_swap
tff(fact_5907_div__div__eq__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_5908_div__div__eq__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_5909_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_5910_dvd__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)),A2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_5911_dvd__mult__imp__div,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2))) ) ).

% dvd_mult_imp_div
tff(fact_5912_dvd__mult__imp__div,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2))) ) ).

% dvd_mult_imp_div
tff(fact_5913_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A2: nat,D2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),C2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5914_div__mult__div__if__dvd,axiom,
    ! [B2: int,A2: int,D2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),C2))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5915_div__power,axiom,
    ! [B2: code_integer,A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B2),A2))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),N) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ) ) ).

% div_power
tff(fact_5916_div__power,axiom,
    ! [B2: nat,A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ) ) ).

% div_power
tff(fact_5917_div__power,axiom,
    ! [B2: int,A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),N) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)) ) ) ).

% div_power
tff(fact_5918_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% le_imp_power_dvd
tff(fact_5919_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% le_imp_power_dvd
tff(fact_5920_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ).

% le_imp_power_dvd
tff(fact_5921_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N))) ) ).

% le_imp_power_dvd
tff(fact_5922_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% le_imp_power_dvd
tff(fact_5923_power__le__dvd,axiom,
    ! [A2: nat,N: nat,B2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_5924_power__le__dvd,axiom,
    ! [A2: real,N: nat,B2: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_5925_power__le__dvd,axiom,
    ! [A2: int,N: nat,B2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_5926_power__le__dvd,axiom,
    ! [A2: complex,N: nat,B2: complex,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_5927_power__le__dvd,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer,M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),B2)) ) ) ).

% power_le_dvd
tff(fact_5928_dvd__power__le,axiom,
    ! [X2: nat,Y3: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y3),M))) ) ) ).

% dvd_power_le
tff(fact_5929_dvd__power__le,axiom,
    ! [X2: real,Y3: real,N: nat,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),M))) ) ) ).

% dvd_power_le
tff(fact_5930_dvd__power__le,axiom,
    ! [X2: int,Y3: int,N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),M))) ) ) ).

% dvd_power_le
tff(fact_5931_dvd__power__le,axiom,
    ! [X2: complex,Y3: complex,N: nat,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),M))) ) ) ).

% dvd_power_le
tff(fact_5932_dvd__power__le,axiom,
    ! [X2: code_integer,Y3: code_integer,N: nat,M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),Y3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),M))) ) ) ).

% dvd_power_le
tff(fact_5933_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_5934_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_5935_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_5936_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_5937_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_5938_return__cons__rule,axiom,
    ! [P: assn,Q: fun_VEBT_VEBTi_assn,X2: vEBT_VEBTi] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_VEBT_VEBTi_assn(Q,X2)))
     => hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(X2),Q) ) ).

% return_cons_rule
tff(fact_5939_return__cons__rule,axiom,
    ! [P: assn,Q: fun_bool_assn,X2: bool] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_bool_assn(Q,X2)))
     => hoare_2774773592644289642e_bool(P,heap_T5885414692442464636n_bool(X2),Q) ) ).

% return_cons_rule
tff(fact_5940_return__cons__rule,axiom,
    ! [P: assn,Q: fun_option_nat_assn,X2: option_nat] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_option_nat_assn(Q,X2)))
     => hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(X2),Q) ) ).

% return_cons_rule
tff(fact_5941_return__cons__rule,axiom,
    ! [P: assn,Q: fun_nat_assn,X2: nat] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_nat_assn(Q,X2)))
     => hoare_3067605981109127869le_nat(P,heap_Time_return_nat(X2),Q) ) ).

% return_cons_rule
tff(fact_5942_even__unset__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5943_even__unset__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5944_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_5945_even__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_5946_even__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_5947_unity__coeff__ex,axiom,
    ! [P: fun_complex_bool,L: complex] :
      ( ? [X3: complex] : pp(aa_complex_bool(P,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,L),X3)))
    <=> ? [X3: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X3),zero_zero_complex)))
          & pp(aa_complex_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_5948_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X3: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X3)))
    <=> ? [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),zero_zero_real)))
          & pp(aa_real_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_5949_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X3: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X3)))
    <=> ? [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X3),zero_zero_rat)))
          & pp(aa_rat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_5950_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X3: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3)))
    <=> ? [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_5951_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X3: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3)))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int)))
          & pp(aa_int_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_5952_unit__dvdE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ~ ( ( A2 != zero_zero_nat )
         => ! [C3: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C3) ) ) ).

% unit_dvdE
tff(fact_5953_unit__dvdE,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ~ ( ( A2 != zero_zero_int )
         => ! [C3: int] : B2 != aa_int_int(aa_int_fun_int_int(times_times_int,A2),C3) ) ) ).

% unit_dvdE
tff(fact_5954_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat )
      <=> ( A2 = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5955_unit__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5956_dvd__div__eq__mult,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = C2 )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5957_dvd__div__eq__mult,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = C2 )
        <=> ( B2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5958_div__dvd__iff__mult,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5959_div__dvd__iff__mult,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5960_dvd__div__iff__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),B2)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5961_dvd__div__iff__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( C2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),B2)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5962_dvd__div__div__eq__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),D2))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D2),C2) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5963_dvd__div__div__eq__mult,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),D2))
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D2),C2) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5964_inf__period_I3_J,axiom,
    ! [D2: real,D: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),D))
     => ! [X6: real,K5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))),T))) ) ) ).

% inf_period(3)
tff(fact_5965_inf__period_I3_J,axiom,
    ! [D2: rat,D: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),D))
     => ! [X6: rat,K5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X6),aa_rat_rat(times_times_rat(K5),D))),T))) ) ) ).

% inf_period(3)
tff(fact_5966_inf__period_I3_J,axiom,
    ! [D2: int,D: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int,K5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D))),T))) ) ) ).

% inf_period(3)
tff(fact_5967_inf__period_I4_J,axiom,
    ! [D2: real,D: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),D))
     => ! [X6: real,K5: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X6),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))),T))) ) ) ).

% inf_period(4)
tff(fact_5968_inf__period_I4_J,axiom,
    ! [D2: rat,D: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),D))
     => ! [X6: rat,K5: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X6),T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X6),aa_rat_rat(times_times_rat(K5),D))),T))) ) ) ).

% inf_period(4)
tff(fact_5969_inf__period_I4_J,axiom,
    ! [D2: int,D: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int,K5: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K5),D))),T))) ) ) ).

% inf_period(4)
tff(fact_5970_unit__eq__div1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = C2 )
      <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) ) ) ) ).

% unit_eq_div1
tff(fact_5971_unit__eq__div1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = C2 )
      <=> ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2) ) ) ) ).

% unit_eq_div1
tff(fact_5972_unit__eq__div2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),B2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_5973_unit__eq__div2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( ( A2 = aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),B2) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_5974_div__mult__unit2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_5975_div__mult__unit2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),A2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_5976_unit__div__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),B2) ) ) ).

% unit_div_commute
tff(fact_5977_unit__div__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),B2) ) ) ).

% unit_div_commute
tff(fact_5978_unit__div__mult__swap,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_5979_unit__div__mult__swap,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_5980_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5981_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5982_is__unit__power__iff,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5983_is__unit__power__iff,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5984_is__unit__power__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),one_one_Code_integer))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A2),one_one_Code_integer))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5985_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_5986_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_5987_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_5988_TBOUND__return,axiom,
    ! [X2: nat] : pp(aa_nat_bool(time_TBOUND_nat(heap_Time_return_nat(X2)),one_one_nat)) ).

% TBOUND_return
tff(fact_5989_TBOUND__return,axiom,
    ! [X2: bool] : pp(aa_nat_bool(time_TBOUND_bool(heap_T5885414692442464636n_bool(X2)),one_one_nat)) ).

% TBOUND_return
tff(fact_5990_TBOUND__return,axiom,
    ! [X2: vEBT_VEBTi] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(heap_T3630416162098727440_VEBTi(X2)),one_one_nat)) ).

% TBOUND_return
tff(fact_5991_TBOUND__return,axiom,
    ! [X2: option_nat] : pp(aa_nat_bool(time_T8353473612707095248on_nat(heap_T3487192422709364219on_nat(X2)),one_one_nat)) ).

% TBOUND_return
tff(fact_5992_real__of__nat__div,axiom,
    ! [D2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D2)) ) ) ).

% real_of_nat_div
tff(fact_5993_time__return,axiom,
    ! [X2: nat,H: heap_e7401611519738050253t_unit] : time_time_nat(heap_Time_return_nat(X2),H) = one_one_nat ).

% time_return
tff(fact_5994_time__return,axiom,
    ! [X2: bool,H: heap_e7401611519738050253t_unit] : time_time_bool(heap_T5885414692442464636n_bool(X2),H) = one_one_nat ).

% time_return
tff(fact_5995_time__return,axiom,
    ! [X2: vEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_time_VEBT_VEBTi(heap_T3630416162098727440_VEBTi(X2),H) = one_one_nat ).

% time_return
tff(fact_5996_time__return,axiom,
    ! [X2: option_nat,H: heap_e7401611519738050253t_unit] : time_time_option_nat(heap_T3487192422709364219on_nat(X2),H) = one_one_nat ).

% time_return
tff(fact_5997_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B2)),C2) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),C2)),B2) ).

% assn_aci(10)
tff(fact_5998_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B2),C2)) = aa_assn_assn(times_times_assn(B2),aa_assn_assn(times_times_assn(A2),C2)) ).

% star_aci(3)
tff(fact_5999_star__aci_I2_J,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(times_times_assn(A2),B2) = aa_assn_assn(times_times_assn(B2),A2) ).

% star_aci(2)
tff(fact_6000_star__assoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B2)),C2) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B2),C2)) ).

% star_assoc
tff(fact_6001_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ).

% is_entails
tff(fact_6002_TBOUND__nth,axiom,
    ! [Xs: array_VEBT_VEBTi,I: nat] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(array_nth_VEBT_VEBTi(Xs,I)),one_one_nat)) ).

% TBOUND_nth
tff(fact_6003_TBOUND__nth,axiom,
    ! [Xs: array_bool,I: nat] : pp(aa_nat_bool(time_TBOUND_bool(array_nth_bool(Xs,I)),one_one_nat)) ).

% TBOUND_nth
tff(fact_6004_TBOUND__nth,axiom,
    ! [Xs: array_option_nat,I: nat] : pp(aa_nat_bool(time_T8353473612707095248on_nat(array_nth_option_nat(Xs,I)),one_one_nat)) ).

% TBOUND_nth
tff(fact_6005_TBOUND__nth,axiom,
    ! [Xs: array_nat,I: nat] : pp(aa_nat_bool(time_TBOUND_nat(array_nth_nat(Xs,I)),one_one_nat)) ).

% TBOUND_nth
tff(fact_6006_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat)) ).

% even_zero
tff(fact_6007_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int)) ).

% even_zero
tff(fact_6008_odd__even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% odd_even_add
tff(fact_6009_odd__even__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% odd_even_add
tff(fact_6010_odd__one,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat)) ).

% odd_one
tff(fact_6011_odd__one,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int)) ).

% odd_one
tff(fact_6012_evenE,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ~ ! [B3: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_6013_evenE,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ~ ! [B3: int] : A2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_6014_bit__eq__rec,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6015_bit__eq__rec,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6016_is__unit__div__mult__cancel__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_6017_is__unit__div__mult__cancel__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_6018_is__unit__div__mult__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_6019_is__unit__div__mult__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_6020_is__unitE,axiom,
    ! [A2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),one_one_nat))
     => ~ ( ( A2 != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B3),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A2 )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A2) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6021_is__unitE,axiom,
    ! [A2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),one_one_int))
     => ~ ( ( A2 != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),one_one_int))
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A2 )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A2) != aa_int_int(aa_int_fun_int_int(times_times_int,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6022_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_6023_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_6024_dvd__power__iff,axiom,
    ! [X2: nat,M: nat,N: nat] :
      ( ( X2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_6025_dvd__power__iff,axiom,
    ! [X2: int,M: nat,N: nat] :
      ( ( X2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_6026_dvd__power__iff,axiom,
    ! [X2: code_integer,M: nat,N: nat] :
      ( ( X2 != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)))
      <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),one_one_Code_integer))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_6027_dvd__power,axiom,
    ! [N: nat,X2: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),aa_nat_rat(power_power_rat(X2),N))) ) ).

% dvd_power
tff(fact_6028_dvd__power,axiom,
    ! [N: nat,X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N))) ) ).

% dvd_power
tff(fact_6029_dvd__power,axiom,
    ! [N: nat,X2: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ).

% dvd_power
tff(fact_6030_dvd__power,axiom,
    ! [N: nat,X2: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N))) ) ).

% dvd_power
tff(fact_6031_dvd__power,axiom,
    ! [N: nat,X2: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N))) ) ).

% dvd_power
tff(fact_6032_dvd__power,axiom,
    ! [N: nat,X2: code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_Code_integer ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),N))) ) ).

% dvd_power
tff(fact_6033_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_6034_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_6035_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_6036_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q3)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q3)))) ) ) ) ).

% dvd_minus_add
tff(fact_6037_even__two__times__div__two,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_6038_even__two__times__div__two,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_6039_power__mono__odd,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))) ) ) ).

% power_mono_odd
tff(fact_6040_power__mono__odd,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))) ) ) ).

% power_mono_odd
tff(fact_6041_power__mono__odd,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N))) ) ) ).

% power_mono_odd
tff(fact_6042_power__mono__odd,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N))) ) ) ).

% power_mono_odd
tff(fact_6043_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_6044_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_6045_even__set__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6046_even__set__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6047_return__sp__rule,axiom,
    ! [P: assn,X2: vEBT_VEBTi] : hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(X2),aa_VEB1545114916007340237i_assn(aTP_Lamm_dw(P),X2)) ).

% return_sp_rule
tff(fact_6048_return__sp__rule,axiom,
    ! [P: assn,X2: bool] : hoare_2774773592644289642e_bool(P,heap_T5885414692442464636n_bool(X2),aa_boo2938360491098410167l_assn(aTP_Lamm_dx(P),X2)) ).

% return_sp_rule
tff(fact_6049_return__sp__rule,axiom,
    ! [P: assn,X2: option_nat] : hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(X2),aa_opt3861602675778688675t_assn(aTP_Lamm_dy(P),X2)) ).

% return_sp_rule
tff(fact_6050_return__sp__rule,axiom,
    ! [P: assn,X2: nat] : hoare_3067605981109127869le_nat(P,heap_Time_return_nat(X2),aa_nat_fun_nat_assn(aTP_Lamm_dz(P),X2)) ).

% return_sp_rule
tff(fact_6051_oddE,axiom,
    ! [A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ~ ! [B3: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_6052_oddE,axiom,
    ! [A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ~ ! [B3: int] : A2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_6053_zero__le__power__eq,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ) ) ).

% zero_le_power_eq
tff(fact_6054_zero__le__power__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ) ) ).

% zero_le_power_eq
tff(fact_6055_zero__le__power__eq,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ) ).

% zero_le_power_eq
tff(fact_6056_zero__le__power__eq,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ) ) ).

% zero_le_power_eq
tff(fact_6057_zero__le__odd__power,axiom,
    ! [N: nat,A2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ) ).

% zero_le_odd_power
tff(fact_6058_zero__le__odd__power,axiom,
    ! [N: nat,A2: code_integer] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ) ).

% zero_le_odd_power
tff(fact_6059_zero__le__odd__power,axiom,
    ! [N: nat,A2: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ).

% zero_le_odd_power
tff(fact_6060_zero__le__odd__power,axiom,
    ! [N: nat,A2: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ) ).

% zero_le_odd_power
tff(fact_6061_zero__le__even__power,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% zero_le_even_power
tff(fact_6062_zero__le__even__power,axiom,
    ! [N: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% zero_le_even_power
tff(fact_6063_zero__le__even__power,axiom,
    ! [N: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))) ) ).

% zero_le_even_power
tff(fact_6064_zero__le__even__power,axiom,
    ! [N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ).

% zero_le_even_power
tff(fact_6065_zero__less__power__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ) ) ).

% zero_less_power_eq
tff(fact_6066_zero__less__power__eq,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A2 != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ) ) ).

% zero_less_power_eq
tff(fact_6067_zero__less__power__eq,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A2 != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ) ) ).

% zero_less_power_eq
tff(fact_6068_zero__less__power__eq,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A2 != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ) ) ).

% zero_less_power_eq
tff(fact_6069_fr__rot,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(A),B)),C))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(B),A)),C)) ) ).

% fr_rot
tff(fact_6070_fr__refl,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),B))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(A),C)),aa_assn_assn(times_times_assn(B),C))) ) ).

% fr_refl
tff(fact_6071_fr__rot__rhs,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(times_times_assn(B),C)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(times_times_assn(C),B))) ) ).

% fr_rot_rhs
tff(fact_6072_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F3: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(times_times_assn(P),F3)))
       => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(R),F3)),Q))
         => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),Q)) ) ) ) ).

% ent_frame_fwd
tff(fact_6073_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% norm_assertion_simps(2)
tff(fact_6074_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% norm_assertion_simps(1)
tff(fact_6075_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_6076_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_6077_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_6078_power__le__zero__eq,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6079_power__le__zero__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),zero_z3403309356797280102nteger))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6080_power__le__zero__eq,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6081_power__le__zero__eq,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6082_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_6083_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_6084_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_6085_frame__rule__left,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(R),P),C2,aa_ass4389913640333755968i_assn(aTP_Lamm_ea(Q),R)) ) ).

% frame_rule_left
tff(fact_6086_frame__rule__left,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bool_assn,R: assn] :
      ( hoare_2774773592644289642e_bool(P,C2,Q)
     => hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(R),P),C2,aa_ass2376637134808068204l_assn(aTP_Lamm_eb(Q),R)) ) ).

% frame_rule_left
tff(fact_6087_frame__rule__left,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(R),P),C2,aa_ass4112278883933122603t_assn(aTP_Lamm_ec(Q),R)) ) ).

% frame_rule_left
tff(fact_6088_frame__rule__left,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(R),P),C2,aa_assn_fun_nat_assn(aTP_Lamm_ed(Q),R)) ) ).

% frame_rule_left
tff(fact_6089_Bernoulli__inequality__even,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_6090_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_6091_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6092_even__mult__exp__div__exp__iff,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6093_even__mult__exp__div__exp__iff,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6094_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_6095_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_6096_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_6097_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ).

% VEBT_internal.Tb.simps(3)
tff(fact_6098_mod__frame__fwd,axiom,
    ! [Ps: assn,H: produc3658429121746597890et_nat,P: assn,R: assn,F3: assn] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(Ps),H))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R))
       => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(times_times_assn(P),F3)))
         => pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(R),F3)),H)) ) ) ) ).

% mod_frame_fwd
tff(fact_6099_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_6100_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat)))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_6101_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_6102_VEBT__internal_OTb_Oelims,axiom,
    ! [X2: nat,Y3: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                     => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_6103_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat)))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_6104_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
         => ~ ! [Va3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_6105_pow__divides__pow__iff,axiom,
    ! [N: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),B2)) ) ) ).

% pow_divides_pow_iff
tff(fact_6106_pow__divides__pow__iff,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),B2)) ) ) ).

% pow_divides_pow_iff
tff(fact_6107_div2__even__ext__nat,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y3)) )
       => ( X2 = Y3 ) ) ) ).

% div2_even_ext_nat
tff(fact_6108_lowi__def,axiom,
    ! [X2: nat,N: nat] : vEBT_VEBT_lowi(X2,N) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% lowi_def
tff(fact_6109_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D4: nat,X4: nat,Y4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y4)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_6110_low__def,axiom,
    ! [X2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_6111_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_6112_mod__mod__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mod_trivial
tff(fact_6113_mod__mod__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mod_trivial
tff(fact_6114_mod__mod__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mod_trivial
tff(fact_6115_mod__add__self2,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_add_self2
tff(fact_6116_mod__add__self2,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_add_self2
tff(fact_6117_mod__add__self2,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_add_self2
tff(fact_6118_mod__add__self1,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_add_self1
tff(fact_6119_mod__add__self1,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_add_self1
tff(fact_6120_mod__add__self1,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_add_self1
tff(fact_6121_minus__mod__self2,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% minus_mod_self2
tff(fact_6122_minus__mod__self2,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% minus_mod_self2
tff(fact_6123_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_6124_nat__mod__eq_H,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = A2 ) ) ).

% nat_mod_eq'
tff(fact_6125_mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_6126_mod__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_6127_mod__by__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_6128_bits__mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_6129_bits__mod__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_6130_bits__mod__by__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_6131_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),B2) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_6132_mod__mult__self1__is__0,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),B2) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_6133_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),A2)),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_6134_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),B2) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_6135_mod__mult__self2__is__0,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),B2) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_6136_mod__mult__self2__is__0,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_6137_bits__mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_6138_bits__mod__div__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_6139_bits__mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_6140_mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = zero_zero_nat ).

% mod_div_trivial
tff(fact_6141_mod__div__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) = zero_zero_int ).

% mod_div_trivial
tff(fact_6142_mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_6143_mod__mult__self1,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self1
tff(fact_6144_mod__mult__self1,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self1
tff(fact_6145_mod__mult__self1,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C2),B2))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self1
tff(fact_6146_mod__mult__self2,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self2
tff(fact_6147_mod__mult__self2,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self2
tff(fact_6148_mod__mult__self2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),C2))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self2
tff(fact_6149_mod__mult__self3,axiom,
    ! [C2: nat,B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self3
tff(fact_6150_mod__mult__self3,axiom,
    ! [C2: int,B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self3
tff(fact_6151_mod__mult__self3,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C2),B2)),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self3
tff(fact_6152_mod__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self4
tff(fact_6153_mod__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self4
tff(fact_6154_mod__mult__self4,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),C2)),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self4
tff(fact_6155_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_6156_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_6157_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_6158_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_6159_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_6160_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_6161_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_6162_bits__one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_6163_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_6164_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_6165_one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_6166_even__mod__2__iff,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)) ) ).

% even_mod_2_iff
tff(fact_6167_even__mod__2__iff,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) ) ).

% even_mod_2_iff
tff(fact_6168_even__mod__2__iff,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)) ) ).

% even_mod_2_iff
tff(fact_6169_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_6170_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_6171_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_6172_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_6173_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_6174_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_6175_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_6176_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_6177_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_6178_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_6179_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_6180_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_6181_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_6182_even__succ__mod__exp,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6183_even__succ__mod__exp,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6184_even__succ__mod__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6185_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mod
tff(fact_6186_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mod
tff(fact_6187_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mod
tff(fact_6188_mod__add__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ).

% mod_add_right_eq
tff(fact_6189_mod__add__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ).

% mod_add_right_eq
tff(fact_6190_mod__add__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C2) ).

% mod_add_right_eq
tff(fact_6191_mod__add__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ).

% mod_add_left_eq
tff(fact_6192_mod__add__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ).

% mod_add_left_eq
tff(fact_6193_mod__add__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C2) ).

% mod_add_left_eq
tff(fact_6194_mod__add__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)),C2) ) ) ) ).

% mod_add_cong
tff(fact_6195_mod__add__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C2) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C2) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B4)),C2) ) ) ) ).

% mod_add_cong
tff(fact_6196_mod__add__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),C2) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B4),C2) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A4),B4)),C2) ) ) ) ).

% mod_add_cong
tff(fact_6197_mod__add__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ).

% mod_add_eq
tff(fact_6198_mod__add__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ).

% mod_add_eq
tff(fact_6199_mod__add__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C2) ).

% mod_add_eq
tff(fact_6200_mod__mult__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ).

% mod_mult_right_eq
tff(fact_6201_mod__mult__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) ).

% mod_mult_right_eq
tff(fact_6202_mod__mult__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),C2) ).

% mod_mult_right_eq
tff(fact_6203_mod__mult__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ).

% mod_mult_left_eq
tff(fact_6204_mod__mult__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) ).

% mod_mult_left_eq
tff(fact_6205_mod__mult__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),C2) ).

% mod_mult_left_eq
tff(fact_6206_mult__mod__right,axiom,
    ! [C2: nat,A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ).

% mult_mod_right
tff(fact_6207_mult__mod__right,axiom,
    ! [C2: int,A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B2)) ).

% mult_mod_right
tff(fact_6208_mult__mod__right,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C2),A2)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C2),B2)) ).

% mult_mod_right
tff(fact_6209_mod__mult__mult2,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),C2) ).

% mod_mult_mult2
tff(fact_6210_mod__mult__mult2,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),C2) ).

% mod_mult_mult2
tff(fact_6211_mod__mult__mult2,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),C2)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),C2)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),C2) ).

% mod_mult_mult2
tff(fact_6212_mod__mult__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),B4)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_6213_mod__mult__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C2) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C2) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A4),B4)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_6214_mod__mult__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),C2) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B4),C2) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A4),B4)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_6215_mod__mult__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ).

% mod_mult_eq
tff(fact_6216_mod__mult__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) ).

% mod_mult_eq
tff(fact_6217_mod__mult__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),C2) ).

% mod_mult_eq
tff(fact_6218_mod__diff__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) ).

% mod_diff_right_eq
tff(fact_6219_mod__diff__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C2) ).

% mod_diff_right_eq
tff(fact_6220_mod__diff__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) ).

% mod_diff_left_eq
tff(fact_6221_mod__diff__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C2) ).

% mod_diff_left_eq
tff(fact_6222_mod__diff__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C2) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C2) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B4)),C2) ) ) ) ).

% mod_diff_cong
tff(fact_6223_mod__diff__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),C2) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B4),C2) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A4),B4)),C2) ) ) ) ).

% mod_diff_cong
tff(fact_6224_mod__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C2) ).

% mod_diff_eq
tff(fact_6225_mod__diff__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C2) ).

% mod_diff_eq
tff(fact_6226_power__mod,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),N)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),B2) ).

% power_mod
tff(fact_6227_power__mod,axiom,
    ! [A2: int,B2: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),N)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),B2) ).

% power_mod
tff(fact_6228_power__mod,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),N)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),B2) ).

% power_mod
tff(fact_6229_mod__mod__cancel,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) ) ) ).

% mod_mod_cancel
tff(fact_6230_mod__mod__cancel,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) ) ) ).

% mod_mod_cancel
tff(fact_6231_mod__mod__cancel,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C2),B2))
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2) ) ) ).

% mod_mod_cancel
tff(fact_6232_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) ) ).

% dvd_mod
tff(fact_6233_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),N))) ) ) ).

% dvd_mod
tff(fact_6234_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(M),N))) ) ) ).

% dvd_mod
tff(fact_6235_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_6236_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_6237_nat__mod__eq,axiom,
    ! [B2: nat,N: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),N))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_6238_mod__plus__right,axiom,
    ! [A2: nat,X2: nat,M: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),X2)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),X2)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),M) ) ) ).

% mod_plus_right
tff(fact_6239_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_6240_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_6241_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),zero_zero_nat))
     => ( ! [M5: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M5),N3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),N3)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_6242_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),A2)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6243_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),A2)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6244_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),A2)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6245_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6246_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6247_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6248_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_6249_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_6250_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Q3)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_6251_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_6252_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_6253_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_6254_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = A2 )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6255_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = A2 )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6256_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = A2 )
    <=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6257_mod__eqE,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) )
     => ~ ! [D4: int] : B2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D4)) ) ).

% mod_eqE
tff(fact_6258_mod__eqE,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2) )
     => ~ ! [D4: code_integer] : B2 != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C2),D4)) ) ).

% mod_eqE
tff(fact_6259_div__add1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2)) ).

% div_add1_eq
tff(fact_6260_div__add1__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2)) ).

% div_add1_eq
tff(fact_6261_div__add1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C2)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C2))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2)) ).

% div_add1_eq
tff(fact_6262_mod__eq__dvd__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))) ) ).

% mod_eq_dvd_iff
tff(fact_6263_mod__eq__dvd__iff,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2))) ) ).

% mod_eq_dvd_iff
tff(fact_6264_dvd__minus__mod,axiom,
    ! [B2: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)))) ).

% dvd_minus_mod
tff(fact_6265_dvd__minus__mod,axiom,
    ! [B2: int,A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)))) ).

% dvd_minus_mod
tff(fact_6266_dvd__minus__mod,axiom,
    ! [B2: code_integer,A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)))) ).

% dvd_minus_mod
tff(fact_6267_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_6268_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P2: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P2))
         => ( ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),P2))
               => ( pp(aa_nat_bool(P,N3))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N3)),P2))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_6269_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_6270_nat__mod__lem,axiom,
    ! [N: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),N))
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_6271_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_6272_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),L))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D2),N)),L) = N ) ) ) ).

% word_rot_lem
tff(fact_6273_nat__minus__mod,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_6274_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_6275_mod__nat__sub,axiom,
    ! [X2: nat,Z: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Y3)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Y3) ) ) ).

% mod_nat_sub
tff(fact_6276_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_6277_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D2) = zero_zero_nat )
     => ? [Q4: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D2),Q4) ) ).

% mod_eq_0D
tff(fact_6278_nat__minus__mod__plus__right,axiom,
    ! [N: nat,X2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),M) ).

% nat_minus_mod_plus_right
tff(fact_6279_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_6280_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) ).

% msrevs(2)
tff(fact_6281_nat__mod__eq__iff,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y3),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_6282_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_6283_zdvd__period,axiom,
    ! [A2: int,D2: int,X2: int,T: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),D2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D2))),T))) ) ) ).

% zdvd_period
tff(fact_6284_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_6285_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6286_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6287_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6288_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6289_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6290_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6291_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6292_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6293_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6294_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_6295_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_6296_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_6297_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_6298_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_6299_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_6300_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_6301_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_6302_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_6303_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6304_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6305_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6306_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6307_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6308_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6309_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6310_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6311_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Q3)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6312_mult__div__mod__eq,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_6313_mult__div__mod__eq,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_6314_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_6315_mod__mult__div__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_6316_mod__mult__div__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_6317_mod__mult__div__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_6318_mod__div__mult__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_6319_mod__div__mult__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_6320_mod__div__mult__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_6321_div__mult__mod__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_6322_div__mult__mod__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_6323_div__mult__mod__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_6324_mod__div__decomp,axiom,
    ! [A2: nat,B2: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ).

% mod_div_decomp
tff(fact_6325_mod__div__decomp,axiom,
    ! [A2: int,B2: int] : A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% mod_div_decomp
tff(fact_6326_mod__div__decomp,axiom,
    ! [A2: code_integer,B2: code_integer] : A2 = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ).

% mod_div_decomp
tff(fact_6327_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) ).

% cancel_div_mod_rules(1)
tff(fact_6328_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2) ).

% cancel_div_mod_rules(1)
tff(fact_6329_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))),C2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C2) ).

% cancel_div_mod_rules(1)
tff(fact_6330_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) ).

% cancel_div_mod_rules(2)
tff(fact_6331_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C2) ).

% cancel_div_mod_rules(2)
tff(fact_6332_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))),C2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C2) ).

% cancel_div_mod_rules(2)
tff(fact_6333_div__mult1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2)) ).

% div_mult1_eq
tff(fact_6334_div__mult1__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2))),C2)) ).

% div_mult1_eq
tff(fact_6335_div__mult1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)),C2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C2))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C2))),C2)) ).

% div_mult1_eq
tff(fact_6336_minus__mult__div__eq__mod,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_6337_minus__mult__div__eq__mod,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_6338_minus__mult__div__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_6339_minus__mod__eq__mult__div,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_6340_minus__mod__eq__mult__div,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_6341_minus__mod__eq__mult__div,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_6342_minus__mod__eq__div__mult,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_6343_minus__mod__eq__div__mult,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_6344_minus__mod__eq__div__mult,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_6345_minus__div__mult__eq__mod,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_6346_minus__div__mult__eq__mod,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_6347_minus__div__mult__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_6348_zmde,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% zmde
tff(fact_6349_zmde,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ).

% zmde
tff(fact_6350_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_6351_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_6352_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B2),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_6353_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_6354_div__less__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),N))) ) ) ) ) ).

% div_less_mono
tff(fact_6355_mod__nat__add,axiom,
    ! [X2: nat,Z: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),Z))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),Z))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),Z))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y3)),Z) ) ) ) ) ) ).

% mod_nat_add
tff(fact_6356_nat__mod__eq__lemma,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y3),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2))
       => ? [Q4: nat] : X2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_6357_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_6358_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_6359_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_6360_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod'_nat_def
tff(fact_6361_div__mod__decomp,axiom,
    ! [A: nat,N: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N)) ).

% div_mod_decomp
tff(fact_6362_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_6363_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)) ).

% modulo_nat_def
tff(fact_6364_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_6365_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_6366_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : vEBT_VEBT_minNulli(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) = heap_T5885414692442464636n_bool(fFalse) ).

% VEBT_internal.minNulli.simps(5)
tff(fact_6367_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] : vEBT_VEBT_minNulli(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux),Uy)) = heap_T5885414692442464636n_bool(fTrue) ).

% VEBT_internal.minNulli.simps(4)
tff(fact_6368_prod__decode__aux_Ocases,axiom,
    ! [X2: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] : X2 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M5) ).

% prod_decode_aux.cases
tff(fact_6369_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6370_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6371_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6372_mod__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_6373_mod__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_6374_mod__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_6375_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_6376_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% field_char_0_class.of_nat_div
tff(fact_6377_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% field_char_0_class.of_nat_div
tff(fact_6378_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% field_char_0_class.of_nat_div
tff(fact_6379_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_6380_mod__lemma,axiom,
    ! [C2: nat,R2: nat,B2: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q3),C2))),R2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))) ) ) ).

% mod_lemma
tff(fact_6381_divmod__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N))) ).

% divmod_def
tff(fact_6382_divmod__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod_def
tff(fact_6383_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_6384_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_6385_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),D2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D2),B2))) ) ) ).

% diff_mod_le
tff(fact_6386_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V739175172307565963ildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(vEBT_Leafi(fFalse,fFalse)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_6387_vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_vebt_buildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(vEBT_Leafi(fFalse,fFalse)) ).

% vebt_buildupi.simps(2)
tff(fact_6388_real__of__nat__div__aux,axiom,
    ! [X2: nat,D2: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(semiri5074537144036343181t_real,D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),D2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),D2))),aa_nat_real(semiri5074537144036343181t_real,D2))) ).

% real_of_nat_div_aux
tff(fact_6389_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : vEBT_vebt_minti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_minti.simps(2)
tff(fact_6390_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : vEBT_vebt_maxti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_maxti.simps(2)
tff(fact_6391_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),M))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M)),one_one_int) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M) = aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M) ) ) ) ) ).

% int_div_sub_1
tff(fact_6392_bset_I9_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T))) ) ) ) ).

% bset(9)
tff(fact_6393_bset_I10_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T))) ) ) ) ).

% bset(10)
tff(fact_6394_aset_I9_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T))) ) ) ) ).

% aset(9)
tff(fact_6395_aset_I10_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A))
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T))) ) ) ) ).

% aset(10)
tff(fact_6396_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2))
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),C2)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C2))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6397_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6398_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6399_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6400_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6401_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6402_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6403_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6404_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6405_even__iff__mod__2__eq__zero,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6406_even__iff__mod__2__eq__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6407_even__iff__mod__2__eq__zero,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6408_odd__iff__mod__2__eq__one,axiom,
    ! [A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6409_odd__iff__mod__2__eq__one,axiom,
    ! [A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6410_odd__iff__mod__2__eq__one,axiom,
    ! [A2: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6411_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_6412_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_6413_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).

% even_diff_iff
tff(fact_6414_VEBT__internal_OminNulli_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y3: heap_Time_Heap_bool] :
      ( ( vEBT_VEBT_minNulli(X2) = Y3 )
     => ( ( ( X2 = vEBT_Leafi(fFalse,fFalse) )
         => ( Y3 != heap_T5885414692442464636n_bool(fTrue) ) )
       => ( ( ? [Uv2: bool] : X2 = vEBT_Leafi(fTrue,Uv2)
           => ( Y3 != heap_T5885414692442464636n_bool(fFalse) ) )
         => ( ( ? [Uu2: bool] : X2 = vEBT_Leafi(Uu2,fTrue)
             => ( Y3 != heap_T5885414692442464636n_bool(fFalse) ) )
           => ( ( ? [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ( Y3 != heap_T5885414692442464636n_bool(fTrue) ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2)
                 => ( Y3 != heap_T5885414692442464636n_bool(fFalse) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
tff(fact_6415_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : vEBT_vebt_minti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = heap_T3487192422709364219on_nat(some_nat(Mi)) ).

% vebt_minti.simps(3)
tff(fact_6416_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : vEBT_vebt_maxti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = heap_T3487192422709364219on_nat(some_nat(Ma)) ).

% vebt_maxti.simps(3)
tff(fact_6417_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6418_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6419_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2))
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6420_bits__stable__imp__add__self,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_6421_bits__stable__imp__add__self,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_6422_bits__stable__imp__add__self,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A2 )
     => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_6423_parity__cases,axiom,
    ! [A2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_6424_parity__cases,axiom,
    ! [A2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_6425_parity__cases,axiom,
    ! [A2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
         => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_6426_mod2__eq__if,axiom,
    ! [A2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_6427_mod2__eq__if,axiom,
    ! [A2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_6428_mod2__eq__if,axiom,
    ! [A2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2))
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_6429_div__exp__mod__exp__eq,axiom,
    ! [A2: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_6430_div__exp__mod__exp__eq,axiom,
    ! [A2: int,N: nat,M: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_6431_div__exp__mod__exp__eq,axiom,
    ! [A2: code_integer,N: nat,M: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_6432_power__mod__div,axiom,
    ! [X2: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ).

% power_mod_div
tff(fact_6433_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),N))) ) ) ).

% verit_le_mono_div
tff(fact_6434_vebt__maxti_Osimps_I1_J,axiom,
    ! [B2: bool,A2: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_maxti(vEBT_Leafi(A2,B2)) = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_maxti(vEBT_Leafi(A2,B2)) = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_maxti(vEBT_Leafi(A2,B2)) = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) ).

% vebt_maxti.simps(1)
tff(fact_6435_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_minti(vEBT_Leafi(A2,B2)) = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
      & ( ~ pp(A2)
       => ( ( pp(B2)
           => ( vEBT_vebt_minti(vEBT_Leafi(A2,B2)) = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
          & ( ~ pp(B2)
           => ( vEBT_vebt_minti(vEBT_Leafi(A2,B2)) = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) ).

% vebt_minti.simps(1)
tff(fact_6436_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_6437_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6438_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6439_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2))
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6440_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6441_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6442_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6443_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N3: nat] : X2 != aa_nat_nat(suc,N3) ) ).

% list_decode.cases
tff(fact_6444_dvd__productE,axiom,
    ! [P2: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y4) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X4),A2))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y4),B2)) ) ) ) ).

% dvd_productE
tff(fact_6445_dvd__productE,axiom,
    ! [P2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
     => ~ ! [X4: int,Y4: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y4) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X4),A2))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y4),B2)) ) ) ) ).

% dvd_productE
tff(fact_6446_division__decomp,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)))
     => ? [B6: nat,C5: nat] :
          ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B6),B2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C5),C2)) ) ) ).

% division_decomp
tff(fact_6447_division__decomp,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)))
     => ? [B6: int,C5: int] :
          ( ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C5) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B6),B2))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C5),C2)) ) ) ).

% division_decomp
tff(fact_6448_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) ) ) ) ).

% Euclid_induct
tff(fact_6449_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_6450_mod__double__modulus,axiom,
    ! [M: code_integer,X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
       => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X2),M) )
          | ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X2),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6451_mod__double__modulus,axiom,
    ! [M: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6452_mod__double__modulus,axiom,
    ! [M: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6453_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))))
         => ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6454_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6455_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6456_unset__bit__Suc,axiom,
    ! [N: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6457_unset__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6458_unset__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6459_set__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6460_set__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6461_set__bit__Suc,axiom,
    ! [N: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6462_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_6463_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_6464_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_6465_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))))
         => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6466_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6467_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6468_vebt__minti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y3: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(X2) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = vEBT_Leafi(A3,B3) )
           => ~ ( ( pp(A3)
                 => ( Y3 = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                & ( ~ pp(A3)
                 => ( ( pp(B3)
                     => ( Y3 = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                    & ( ~ pp(B3)
                     => ( Y3 = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y3 != heap_T3487192422709364219on_nat(some_nat(Mi2)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_6469_vebt__maxti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y3: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(X2) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = vEBT_Leafi(A3,B3) )
           => ~ ( ( pp(B3)
                 => ( Y3 = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                & ( ~ pp(B3)
                 => ( ( pp(A3)
                     => ( Y3 = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                    & ( ~ pp(A3)
                     => ( Y3 = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y3 != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y3 != heap_T3487192422709364219on_nat(some_nat(Ma2)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_6470_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_6471_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D4: nat,X4: nat,Y4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A2))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B2))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y4)),D4) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y4)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_6472_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B2))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y3)),D2) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y3)),D2) ) )
         => ? [X4: nat,Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),Y4)),D2) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y4)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_6473_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D4: nat,X4: nat,Y4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A2))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B2))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y4)) = D4 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y4)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_6474_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2))))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_6475_div__half__nat,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( Y3 != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),Y3)) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))),Y3)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))),Y3))),Y3))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))),Y3)))) ) ) ).

% div_half_nat
tff(fact_6476_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_6477_flip__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6478_flip__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6479_flip__bit__Suc,axiom,
    ! [N: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6480_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys))))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys),N) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s3445333598471063425nteger(Ys))),aa_nat_Code_integer(nth_Code_integer(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s3445333598471063425nteger(Ys)))) ) ) ).

% product_nth
tff(fact_6481_product__nth,axiom,
    ! [N: nat,Xs: list_num,Ys: list_num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys))))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),N) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_6482_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys),N) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys)))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_6483_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),N) = produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6484_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys))))
     => ( nth_Pr6842391030413306568T_real(produc4908677263432625371T_real(Xs,Ys),N) = produc8117437818029410057T_real(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6485_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys),N) = produc3035007008238225530T_bool(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_6486_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),N) = produc738532404422230701BT_nat(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_6487_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys))))
     => ( nth_Pr316670251186196177_VEBTi(produc316462671093861988_VEBTi(Xs,Ys),N) = produc6084888613844515218_VEBTi(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6488_product__nth,axiom,
    ! [N: nat,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr5429231388385915574T_VEBT(produc3722688996059531265T_VEBT(Xs,Ys),N) = produc6931449550656315951T_VEBT(aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6489_product__nth,axiom,
    ! [N: nat,Xs: list_real,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys))))
     => ( nth_Pr7489471911767062336l_real(product_real_real(Xs,Ys),N) = produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6490_vebt__assn__raw_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y3: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,X2),Xa) = Y3 )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
           => ! [Ai2: bool,Bi2: bool] :
                ( ( Xa = vEBT_Leafi(Ai2,Bi2) )
               => ( Y3 != aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_bool_bool(fequal_bool(Ai2),A3)),aa_bool_bool(fequal_bool(Bi2),B3))) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                 => ( Y3 != aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,Mmoi),Mmo)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Degi),Deg2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_ee(Tree_list),Tree_array))) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va3,Vb3),Vc3)
             => ( ? [Vd: bool,Ve: bool] : Xa = vEBT_Leafi(Vd,Ve)
               => ( Y3 != bot_bot_assn ) ) )
           => ~ ( ? [Vd: bool,Ve: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Vd),Ve)
               => ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va3,Vb3),Vc3)
                 => ( Y3 != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_6491_ex__assn__const,axiom,
    ! [C2: assn] : ex_ass463751140784270563_VEBTi(aTP_Lamm_ef(C2)) = C2 ).

% ex_assn_const
tff(fact_6492_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_6493_norm__assertion__simps_I17_J,axiom,
    ! [R: assn,Q: fun_li3352006551604695530i_assn] : aa_assn_assn(times_times_assn(R),ex_ass463751140784270563_VEBTi(Q)) = ex_ass463751140784270563_VEBTi(aa_fun6795544665460849810i_assn(aTP_Lamm_eg(R),Q)) ).

% norm_assertion_simps(17)
tff(fact_6494_norm__assertion__simps_I16_J,axiom,
    ! [Q: fun_li3352006551604695530i_assn,R: assn] : aa_assn_assn(times_times_assn(ex_ass463751140784270563_VEBTi(Q)),R) = ex_ass463751140784270563_VEBTi(aa_ass1897699608592665680i_assn(aTP_Lamm_eh(Q),R)) ).

% norm_assertion_simps(16)
tff(fact_6495_triv__exI,axiom,
    ! [Q: fun_li3352006551604695530i_assn,X2: list_VEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_lis2508312872377092907i_assn(Q,X2)),ex_ass463751140784270563_VEBTi(Q))) ).

% triv_exI
tff(fact_6496_mod__ex__dist,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(ex_ass463751140784270563_VEBTi(P)),H))
    <=> ? [X3: list_VEBT_VEBTi] : pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_lis2508312872377092907i_assn(P,X3)),H)) ) ).

% mod_ex_dist
tff(fact_6497_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_6498_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_6499_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_6500_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real] : size_s5035110155006384947T_real(produc4908677263432625371T_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6501_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool] : size_s1343096544515248124T_bool(produc5466675335756156456T_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_6502_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6503_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] : size_s3387956428969716412_VEBTi(produc316462671093861988_VEBTi(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys)) ).

% length_product
tff(fact_6504_length__product,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT] : size_s3289364478449617953T_VEBT(produc3722688996059531265T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_6505_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] : size_s3932428310213730859l_real(product_real_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6506_length__product,axiom,
    ! [Xs: list_real,Ys: list_bool] : size_s2850477280887376644l_bool(product_real_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_6507_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] : size_s1877336372972134351al_nat(product_real_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6508_length__product,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBTi] : size_s6963394564282275252_VEBTi(produc5599769701989005224_VEBTi(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_s7982070591426661849_VEBTi(Ys)) ).

% length_product
tff(fact_6509_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_6510_mod__h__bot__iff_I8_J,axiom,
    ! [R: fun_li3352006551604695530i_assn,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(ex_ass463751140784270563_VEBTi(R)),produc7507926704131184380et_nat(H,bot_bot_set_nat)))
    <=> ? [X3: list_VEBT_VEBTi] : pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_lis2508312872377092907i_assn(R,X3)),produc7507926704131184380et_nat(H,bot_bot_set_nat))) ) ).

% mod_h_bot_iff(8)
tff(fact_6511_one__mod__exp__eq__one,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_6512_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_6513_mod__plus__cong,axiom,
    ! [B2: int,B4: int,X2: int,X5: int,Y3: int,Y6: int,Z5: int] :
      ( ( B2 = B4 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X5),B4) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y3),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y6),B4) )
         => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),Y6) = Z5 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z5),B4) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_6514_zmod__helper,axiom,
    ! [N: int,M: int,K: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),M) = K )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,N),A2)),M) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2)),M) ) ) ).

% zmod_helper
tff(fact_6515_Word_Omod__minus__cong,axiom,
    ! [B2: int,B4: int,X2: int,X5: int,Y3: int,Y6: int,Z5: int] :
      ( ( B2 = B4 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X5),B4) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y3),B4) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y6),B4) )
         => ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),Y6) = Z5 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z5),B4) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_6516_ex__distrib__star,axiom,
    ! [P: fun_li3352006551604695530i_assn,Q: assn] : ex_ass463751140784270563_VEBTi(aa_ass1897699608592665680i_assn(aTP_Lamm_eh(P),Q)) = aa_assn_assn(times_times_assn(ex_ass463751140784270563_VEBTi(P)),Q) ).

% ex_distrib_star
tff(fact_6517_enorm__exI_H,axiom,
    ! [Z6: fun_li5928143645361859307i_bool,P: assn,Q: fun_li3352006551604695530i_assn] :
      ( ! [X4: list_VEBT_VEBTi] :
          ( pp(aa_lis8521252006704567508i_bool(Z6,X4))
         => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_lis2508312872377092907i_assn(Q,X4))) )
     => ( ? [X_13: list_VEBT_VEBTi] : pp(aa_lis8521252006704567508i_bool(Z6,X_13))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),ex_ass463751140784270563_VEBTi(Q))) ) ) ).

% enorm_exI'
tff(fact_6518_ent__ex__preI,axiom,
    ! [P: fun_li3352006551604695530i_assn,Q: assn] :
      ( ! [X4: list_VEBT_VEBTi] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_lis2508312872377092907i_assn(P,X4)),Q))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,ex_ass463751140784270563_VEBTi(P)),Q)) ) ).

% ent_ex_preI
tff(fact_6519_ent__ex__postI,axiom,
    ! [P: assn,Q: fun_li3352006551604695530i_assn,X2: list_VEBT_VEBTi] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_lis2508312872377092907i_assn(Q,X2)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),ex_ass463751140784270563_VEBTi(Q))) ) ).

% ent_ex_postI
tff(fact_6520_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ! [X4: list_VEBT_VEBTi] : hoare_1429296392585015714_VEBTi(aa_lis2508312872377092907i_assn(P,X4),F,Q)
     => hoare_1429296392585015714_VEBTi(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_6521_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( ! [X4: list_VEBT_VEBTi] : hoare_2774773592644289642e_bool(aa_lis2508312872377092907i_assn(P,X4),F,Q)
     => hoare_2774773592644289642e_bool(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_6522_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ! [X4: list_VEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(P,X4),F,Q)
     => hoare_7629718768684598413on_nat(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_6523_norm__pre__ex__rule,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ! [X4: list_VEBT_VEBTi] : hoare_3067605981109127869le_nat(aa_lis2508312872377092907i_assn(P,X4),F,Q)
     => hoare_3067605981109127869le_nat(ex_ass463751140784270563_VEBTi(P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_6524_post__exI__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VE3708348504534315036i_assn,X2: list_VEBT_VEBTi] :
      ( hoare_1429296392585015714_VEBTi(P,C2,aa_lis5319508155010942429i_assn(aTP_Lamm_ei(Q),X2))
     => hoare_1429296392585015714_VEBTi(P,C2,aTP_Lamm_ej(Q)) ) ).

% post_exI_rule
tff(fact_6525_post__exI__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_bool,Q: fun_bo5386306568165246356i_assn,X2: list_VEBT_VEBTi] :
      ( hoare_2774773592644289642e_bool(P,C2,aa_lis822886067112437519l_assn(aTP_Lamm_ek(Q),X2))
     => hoare_2774773592644289642e_bool(P,C2,aTP_Lamm_el(Q)) ) ).

% post_exI_rule
tff(fact_6526_post__exI__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_op8552747401449489031i_assn,X2: list_VEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat(P,C2,aa_lis7387135221810377032t_assn(aTP_Lamm_em(Q),X2))
     => hoare_7629718768684598413on_nat(P,C2,aTP_Lamm_en(Q)) ) ).

% post_exI_rule
tff(fact_6527_post__exI__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_na8005944528675429559i_assn,X2: list_VEBT_VEBTi] :
      ( hoare_3067605981109127869le_nat(P,C2,aa_lis7348694692495225848t_assn(aTP_Lamm_eo(Q),X2))
     => hoare_3067605981109127869le_nat(P,C2,aTP_Lamm_ep(Q)) ) ).

% post_exI_rule
tff(fact_6528_mod__exE,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(ex_ass463751140784270563_VEBTi(P)),H))
     => ~ ! [X4: list_VEBT_VEBTi] : ~ pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_lis2508312872377092907i_assn(P,X4)),H)) ) ).

% mod_exE
tff(fact_6529_mod__exI,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( ? [X6: list_VEBT_VEBTi] : pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_lis2508312872377092907i_assn(P,X6)),H))
     => pp(aa_Pro2991721470636411351t_bool(rep_assn(ex_ass463751140784270563_VEBTi(P)),H)) ) ).

% mod_exI
tff(fact_6530_ex__one__point__gen,axiom,
    ! [P: fun_li3352006551604695530i_assn,V: list_VEBT_VEBTi] :
      ( ! [H2: produc3658429121746597890et_nat,X4: list_VEBT_VEBTi] :
          ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_lis2508312872377092907i_assn(P,X4)),H2))
         => ( X4 = V ) )
     => ( ex_ass463751140784270563_VEBTi(P) = aa_lis2508312872377092907i_assn(P,V) ) ) ).

% ex_one_point_gen
tff(fact_6531_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_li8825114373765265308i_assn,X2: list_VEBT_VEBTi,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,aa_lis5319508155010942429i_assn(Q,X2)),T))
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,aTP_Lamm_er(Q)),T)) ) ).

% norm_post_ex_rule_htt
tff(fact_6532_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_Time_Heap_bool,Q: fun_li4779488348070916006l_assn,X2: list_VEBT_VEBTi,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(P,F,aa_lis822886067112437519l_assn(Q,X2)),T))
     => pp(aa_nat_bool(time_htt_bool(P,F,aTP_Lamm_et(Q)),T)) ) ).

% norm_post_ex_rule_htt
tff(fact_6533_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_T2636463487746394924on_nat,Q: fun_li3309467822378783623t_assn,X2: list_VEBT_VEBTi,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(P,F,aa_lis7387135221810377032t_assn(Q,X2)),T))
     => pp(aa_nat_bool(time_htt_option_nat(P,F,aTP_Lamm_ev(Q)),T)) ) ).

% norm_post_ex_rule_htt
tff(fact_6534_norm__post__ex__rule__htt,axiom,
    ! [P: assn,F: heap_Time_Heap_nat,Q: fun_li7023630697154767927t_assn,X2: list_VEBT_VEBTi,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(P,F,aa_lis7348694692495225848t_assn(Q,X2)),T))
     => pp(aa_nat_bool(time_htt_nat(P,F,aTP_Lamm_ex(Q)),T)) ) ).

% norm_post_ex_rule_htt
tff(fact_6535_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( ! [X4: list_VEBT_VEBTi] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_lis2508312872377092907i_assn(P,X4),F,Q),T))
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(ex_ass463751140784270563_VEBTi(P),F,Q),T)) ) ).

% norm_pre_ex_rule_htt
tff(fact_6536_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( ! [X4: list_VEBT_VEBTi] : pp(aa_nat_bool(time_htt_bool(aa_lis2508312872377092907i_assn(P,X4),F,Q),T))
     => pp(aa_nat_bool(time_htt_bool(ex_ass463751140784270563_VEBTi(P),F,Q),T)) ) ).

% norm_pre_ex_rule_htt
tff(fact_6537_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( ! [X4: list_VEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_lis2508312872377092907i_assn(P,X4),F,Q),T))
     => pp(aa_nat_bool(time_htt_option_nat(ex_ass463751140784270563_VEBTi(P),F,Q),T)) ) ).

% norm_pre_ex_rule_htt
tff(fact_6538_norm__pre__ex__rule__htt,axiom,
    ! [P: fun_li3352006551604695530i_assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( ! [X4: list_VEBT_VEBTi] : pp(aa_nat_bool(time_htt_nat(aa_lis2508312872377092907i_assn(P,X4),F,Q),T))
     => pp(aa_nat_bool(time_htt_nat(ex_ass463751140784270563_VEBTi(P),F,Q),T)) ) ).

% norm_pre_ex_rule_htt
tff(fact_6539_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_6540_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_6541_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).

% neg_mod_bound
tff(fact_6542_int__mod__ge,axiom,
    ! [A2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N))) ) ) ).

% int_mod_ge
tff(fact_6543_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))) ) ) ).

% neg_mod_conj
tff(fact_6544_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_6545_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_6546_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_6547_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_6548_int__mod__lem,axiom,
    ! [N: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),N)) )
      <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) = B2 ) ) ) ).

% int_mod_lem
tff(fact_6549_int__mod__eq,axiom,
    ! [B2: int,N: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),N))
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) )
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_6550_int__mod__le_H,axiom,
    ! [B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),N)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),N))) ) ).

% int_mod_le'
tff(fact_6551_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) ) ) ) ).

% nonneg_mod_div
tff(fact_6552_div__mod__decomp__int,axiom,
    ! [A: int,N: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),N)),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),N)) ).

% div_mod_decomp_int
tff(fact_6553_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% mod_div_equality_div_eq
tff(fact_6554_nat__mod__as__int,axiom,
    ! [X6: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X6),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_mod_as_int
tff(fact_6555_pos__mod__bound2,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% pos_mod_bound2
tff(fact_6556_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_6557_int__mod__ge_H,axiom,
    ! [B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N))) ) ) ).

% int_mod_ge'
tff(fact_6558_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_6559_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_6560_nat__mod__distrib,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y3)) ) ) ) ).

% nat_mod_distrib
tff(fact_6561_real__of__int__div__aux,axiom,
    ! [X2: int,D2: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,X2)),aa_int_real(ring_1_of_int_real,D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),D2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X2),D2))),aa_int_real(ring_1_of_int_real,D2))) ).

% real_of_int_div_aux
tff(fact_6562_pos__mod__sign2,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% pos_mod_sign2
tff(fact_6563_nmod2,axiom,
    ! [N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int )
      | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% nmod2
tff(fact_6564_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_6565_mod__exp__less__eq__exp,axiom,
    ! [A2: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% mod_exp_less_eq_exp
tff(fact_6566_mod__power__lem,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)) = zero_zero_int ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) ) ) ) ) ).

% mod_power_lem
tff(fact_6567_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_6568_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),R2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_6569_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q3)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_6570_mod__add__if__z,axiom,
    ! [X2: int,Z: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
             => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),Z))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3) ) )
                & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),Z))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),Z) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3)),Z) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_6571_mod__sub__if__z,axiom,
    ! [X2: int,Z: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
             => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),Z) = aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3) ) )
                & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),Z) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_6572_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_6573_axxmod2,axiom,
    ! [X2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),X2)),X2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int )
      & ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),X2)),X2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_6574_z1pmod2,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% z1pmod2
tff(fact_6575_verit__le__mono__div__int,axiom,
    ! [A: int,B: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),N)),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),N)),zero_zero_int)),one_one_int),zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_6576_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_6577_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_6578_p1mod22k,axiom,
    ! [B2: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),one_one_int) ).

% p1mod22k
tff(fact_6579_p1mod22k_H,axiom,
    ! [B2: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ).

% p1mod22k'
tff(fact_6580_eq__diff__eq_H,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( X2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),Z) )
    <=> ( Y3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Z) ) ) ).

% eq_diff_eq'
tff(fact_6581_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_6582_eme1p,axiom,
    ! [N: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)),D2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),D2)) ) ) ) ) ).

% eme1p
tff(fact_6583_emep1,axiom,
    ! [N: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int)),D2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),D2)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_6584_even__flip__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A2)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6585_even__flip__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A2)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6586_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K))))) ) ).

% sb_inc_lem
tff(fact_6587_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg,Tree_list2),Summary)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2)) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,Mmoi2),Mmo2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Degi2),Deg)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_ee(Tree_list2),Tree_array2))) ).

% vebt_assn_raw.simps(2)
tff(fact_6588_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A2))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_6589_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_6590_obtain__set__pred,axiom,
    ! [Z: nat,X2: nat,A: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(A),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A))
         => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A),X2),X_12)) ) ) ) ).

% obtain_set_pred
tff(fact_6591_obtain__set__succ,axiom,
    ! [X2: nat,Z: nat,A: set_nat,B: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(A),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B))
         => ( ( A = B )
           => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A),X2),X_12)) ) ) ) ) ).

% obtain_set_succ
tff(fact_6592_vebt__buildup_Oelims,axiom,
    ! [X2: nat,Y3: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,X2) = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ) )
         => ~ ! [Va3: nat] :
                ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                     => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_6593_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B2: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_nat_real(semiri5074537144036343181t_real,K))),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_6594_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBTi_bool,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBTi_bool(P,X2))
       => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,X2),I))) ) ) ).

% intind
tff(fact_6595_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X2))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X2)),I))) ) ) ).

% intind
tff(fact_6596_intind,axiom,
    ! [I: nat,N: nat,P: fun_bool_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_bool_bool(P,X2))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(replicate_bool(N,X2)),I))) ) ) ).

% intind
tff(fact_6597_intind,axiom,
    ! [I: nat,N: nat,P: fun_real_bool,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_real_bool(P,X2))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(replicate_real(N,X2)),I))) ) ) ).

% intind
tff(fact_6598_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X2))
       => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X2)),I))) ) ) ).

% intind
tff(fact_6599_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_6600_succ__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A2),X_12))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X6: nat] :
              ( pp(aa_set_nat_bool(member_nat(X6),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),X6)) ) ) ) ).

% succ_none_empty
tff(fact_6601_pred__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A2),X_12))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X6: nat] :
              ( pp(aa_set_nat_bool(member_nat(X6),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),A2)) ) ) ) ).

% pred_none_empty
tff(fact_6602_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_T8145700208782473153_VEBTi,A: fun_VE905575822219544865i_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_1429296392585015714_VEBTi(Q,X2,aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_ey(Q),A),Y3))
     => hoare_3904069481286416050_VEBTi(Q,vEBT_V1859673955506687831_VEBTi(N,X2),aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_ez(Q),A),Y3),N)) ) ).

% repli_cons_repl
tff(fact_6603_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_Time_Heap_bool,A: fun_VE1323945509527881313l_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_2774773592644289642e_bool(Q,X2,aa_VEB2879522196611654106l_assn(aa_fun6755658226996116224l_assn(aTP_Lamm_fa(Q),A),Y3))
     => hoare_9026296798553663728t_bool(Q,vEBT_V6044891547498605301i_bool(N,X2),aa_nat2784677144853144906l_assn(aa_VEB3184745993655927913l_assn(aa_fun7303525306808986255l_assn(aTP_Lamm_fb(Q),A),Y3),N)) ) ).

% repli_cons_repl
tff(fact_6604_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_T2636463487746394924on_nat,A: fun_VE6334927249922962060t_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_7629718768684598413on_nat(Q,X2,aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_fc(Q),A),Y3))
     => hoare_6480275734082232733on_nat(Q,vEBT_V792416675989592002on_nat(N,X2),aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_fd(Q),A),Y3),N)) ) ).

% repli_cons_repl
tff(fact_6605_repli__cons__repl,axiom,
    ! [Q: assn,X2: heap_Time_Heap_nat,A: fun_VE4931415397359990460t_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_3067605981109127869le_nat(Q,X2,aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_fe(Q),A),Y3))
     => hoare_7964568885773372237st_nat(Q,vEBT_V7726092123322077554ei_nat(N,X2),aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_ff(Q),A),Y3),N)) ) ).

% repli_cons_repl
tff(fact_6606_repli__emp,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,A: fun_VE905575822219544865i_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_1429296392585015714_VEBTi(one_one_assn,X2,aa_VEB3717855479479287634i_assn(A,Y3))
     => hoare_3904069481286416050_VEBTi(one_one_assn,vEBT_V1859673955506687831_VEBTi(N,X2),vEBT_L6296928887356842470_VEBTi(A,replicate_VEBT_VEBT(N,Y3))) ) ).

% repli_emp
tff(fact_6607_repli__emp,axiom,
    ! [X2: heap_Time_Heap_bool,A: fun_VE1323945509527881313l_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_2774773592644289642e_bool(one_one_assn,X2,aa_VEB2879522196611654106l_assn(A,Y3))
     => hoare_9026296798553663728t_bool(one_one_assn,vEBT_V6044891547498605301i_bool(N,X2),vEBT_L5320321625355440422T_bool(A,replicate_VEBT_VEBT(N,Y3))) ) ).

% repli_emp
tff(fact_6608_repli__emp,axiom,
    ! [X2: heap_T2636463487746394924on_nat,A: fun_VE6334927249922962060t_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_7629718768684598413on_nat(one_one_assn,X2,aa_VEB931861714549014589t_assn(A,Y3))
     => hoare_6480275734082232733on_nat(one_one_assn,vEBT_V792416675989592002on_nat(N,X2),vEBT_L8010285020845282001on_nat(A,replicate_VEBT_VEBT(N,Y3))) ) ).

% repli_emp
tff(fact_6609_repli__emp,axiom,
    ! [X2: heap_Time_Heap_nat,A: fun_VE4931415397359990460t_assn,Y3: vEBT_VEBT,N: nat] :
      ( hoare_3067605981109127869le_nat(one_one_assn,X2,aa_VEB8640158415945897581t_assn(A,Y3))
     => hoare_7964568885773372237st_nat(one_one_assn,vEBT_V7726092123322077554ei_nat(N,X2),vEBT_L8296926524756676353BT_nat(A,replicate_VEBT_VEBT(N,Y3))) ) ).

% repli_emp
tff(fact_6610_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs))) ).

% List.finite_set
tff(fact_6611_List_Ofinite__set,axiom,
    ! [Xs: list_real] : pp(aa_set_real_bool(finite_finite_real,set_real2(Xs))) ).

% List.finite_set
tff(fact_6612_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_6613_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_6614_List_Ofinite__set,axiom,
    ! [Xs: list_Code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_Code_integer2(Xs))) ).

% List.finite_set
tff(fact_6615_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_6616_powr__one__eq__one,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),A2) = one_one_real ).

% powr_one_eq_one
tff(fact_6617_replicate__eq__replicate,axiom,
    ! [M: nat,X2: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X2) = replicate_VEBT_VEBT(N,Y3) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X2 = Y3 ) ) ) ) ).

% replicate_eq_replicate
tff(fact_6618_length__replicate,axiom,
    ! [N: nat,X2: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X2)) = N ).

% length_replicate
tff(fact_6619_length__replicate,axiom,
    ! [N: nat,X2: real] : size_size_list_real(replicate_real(N,X2)) = N ).

% length_replicate
tff(fact_6620_length__replicate,axiom,
    ! [N: nat,X2: bool] : size_size_list_bool(replicate_bool(N,X2)) = N ).

% length_replicate
tff(fact_6621_length__replicate,axiom,
    ! [N: nat,X2: nat] : size_size_list_nat(replicate_nat(N,X2)) = N ).

% length_replicate
tff(fact_6622_length__replicate,axiom,
    ! [N: nat,X2: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(replicate_VEBT_VEBTi(N,X2)) = N ).

% length_replicate
tff(fact_6623_map__replicate,axiom,
    ! [F: fun_VEBT_VEBT_nat,N: nat,X2: vEBT_VEBT] : map_VEBT_VEBT_nat(F,replicate_VEBT_VEBT(N,X2)) = replicate_nat(N,aa_VEBT_VEBT_nat(F,X2)) ).

% map_replicate
tff(fact_6624_map__replicate,axiom,
    ! [F: fun_VEBT_VEBT_real,N: nat,X2: vEBT_VEBT] : map_VEBT_VEBT_real(F,replicate_VEBT_VEBT(N,X2)) = replicate_real(N,aa_VEBT_VEBT_real(F,X2)) ).

% map_replicate
tff(fact_6625_map__replicate,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,N: nat,X2: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,replicate_VEBT_VEBT(N,X2)) = replicate_VEBT_VEBT(N,aa_VEB5832953974749077948T_VEBT(F,X2)) ).

% map_replicate
tff(fact_6626_infinite__Icc__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A2,B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% infinite_Icc_iff
tff(fact_6627_infinite__Icc__iff,axiom,
    ! [A2: real,B2: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A2,B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% infinite_Icc_iff
tff(fact_6628_infinite__Ico__iff,axiom,
    ! [A2: real,B2: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or66887138388493659n_real(A2,B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% infinite_Ico_iff
tff(fact_6629_infinite__Ico__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A2,B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% infinite_Ico_iff
tff(fact_6630_powr__zero__eq__one,axiom,
    ! [X2: real] :
      ( ( ( X2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),zero_zero_real) = zero_zero_real ) )
      & ( ( X2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_6631_Ball__set__replicate,axiom,
    ! [N: nat,A2: real,P: fun_real_bool] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(replicate_real(N,A2))))
         => pp(aa_real_bool(P,X3)) )
    <=> ( pp(aa_real_bool(P,A2))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6632_Ball__set__replicate,axiom,
    ! [N: nat,A2: nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A2))))
         => pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A2))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6633_Ball__set__replicate,axiom,
    ! [N: nat,A2: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A2))))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A2))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6634_Bex__set__replicate,axiom,
    ! [N: nat,A2: real,P: fun_real_bool] :
      ( ? [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(replicate_real(N,A2))))
          & pp(aa_real_bool(P,X3)) )
    <=> ( pp(aa_real_bool(P,A2))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6635_Bex__set__replicate,axiom,
    ! [N: nat,A2: nat,P: fun_nat_bool] :
      ( ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A2))))
          & pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A2))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6636_Bex__set__replicate,axiom,
    ! [N: nat,A2: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A2))))
          & pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A2))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6637_in__set__replicate,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( pp(aa_set_int_bool(member_int(X2),set_int2(replicate_int(N,Y3))))
    <=> ( ( X2 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6638_in__set__replicate,axiom,
    ! [X2: real,N: nat,Y3: real] :
      ( pp(aa_set_real_bool(member_real(X2),set_real2(replicate_real(N,Y3))))
    <=> ( ( X2 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6639_in__set__replicate,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(replicate_nat(N,Y3))))
    <=> ( ( X2 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6640_in__set__replicate,axiom,
    ! [X2: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y3))))
    <=> ( ( X2 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6641_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,X2),I) = X2 ) ) ).

% nth_replicate
tff(fact_6642_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X2)),I) = X2 ) ) ).

% nth_replicate
tff(fact_6643_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(nth_bool(replicate_bool(N,X2)),I))
      <=> pp(X2) ) ) ).

% nth_replicate
tff(fact_6644_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_real(nth_real(replicate_real(N,X2)),I) = X2 ) ) ).

% nth_replicate
tff(fact_6645_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X2)),I) = X2 ) ) ).

% nth_replicate
tff(fact_6646_powr__nonneg__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X2)),zero_zero_real))
    <=> ( A2 = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_6647_powr__less__cancel__iff,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% powr_less_cancel_iff
tff(fact_6648_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_6649_powr__eq__one__iff,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A2))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X2) = one_one_real )
      <=> ( X2 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_6650_powr__one__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),one_one_real) = X2 )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% powr_one_gt_zero_iff
tff(fact_6651_powr__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),one_one_real) = X2 ) ) ).

% powr_one
tff(fact_6652_powr__le__cancel__iff,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ) ).

% powr_le_cancel_iff
tff(fact_6653_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,M)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_powr_numeral_real
tff(fact_6654_set__replicate,axiom,
    ! [N: nat,X2: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X2)) = insert_real(X2,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_6655_set__replicate,axiom,
    ! [N: nat,X2: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X2)) = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_6656_set__replicate,axiom,
    ! [N: nat,X2: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X2)) = insert_nat(X2,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_6657_set__replicate,axiom,
    ! [N: nat,X2: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X2)) = insert_int(X2,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_6658_powr__log__cancel,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)) = X2 ) ) ) ) ).

% powr_log_cancel
tff(fact_6659_log__powr__cancel,axiom,
    ! [A2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Y3)) = Y3 ) ) ) ).

% log_powr_cancel
tff(fact_6660_powr__numeral,axiom,
    ! [X2: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_num_real(numeral_numeral_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,N)) ) ) ).

% powr_numeral
tff(fact_6661_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_6662_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => finite6197958912794628473et_int(collect_set_int(aa_set7694327796961020398t_bool(aTP_Lamm_fg(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6663_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S4: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set8732061614394825996r_bool(aTP_Lamm_fh(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6664_finite__if__eq__beyond__finite,axiom,
    ! [S: set_complex,S4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => finite6551019134538273531omplex(collect_set_complex(aa_set6592632205191297002x_bool(aTP_Lamm_fi(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6665_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => finite1152437895449049373et_nat(collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_fj(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6666_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N6)) ) ).

% bounded_nat_set_is_finite
tff(fact_6667_finite__nat__set__iff__bounded,axiom,
    ! [N6: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N6))
    <=> ? [M7: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M7)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_6668_finite__nat__set__iff__bounded__le,axiom,
    ! [N6: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N6))
    <=> ? [M7: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M7)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_6669_finite__list,axiom,
    ! [A: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A ) ).

% finite_list
tff(fact_6670_finite__list,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ? [Xs2: list_real] : set_real2(Xs2) = A ) ).

% finite_list
tff(fact_6671_finite__list,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A ) ).

% finite_list
tff(fact_6672_finite__list,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ? [Xs2: list_int] : set_int2(Xs2) = A ) ).

% finite_list
tff(fact_6673_finite__list,axiom,
    ! [A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ? [Xs2: list_Code_integer] : set_Code_integer2(Xs2) = A ) ).

% finite_list
tff(fact_6674_finite__list,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A ) ).

% finite_list
tff(fact_6675_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fk(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_6676_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),aa_nat_nat(F,N3)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fl(F),U)))) ) ).

% finite_less_ub
tff(fact_6677_finite__lists__length__eq,axiom,
    ! [A: set_Code_integer,N: nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat3680331675976215733r_bool(aTP_Lamm_fm(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6678_finite__lists__length__eq,axiom,
    ! [A: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_fn(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6679_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_fo(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6680_finite__lists__length__eq,axiom,
    ! [A: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_fp(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6681_finite__lists__length__eq,axiom,
    ! [A: set_bool,N: nat] :
      ( finite_finite_bool(A)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fq(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6682_finite__lists__length__eq,axiom,
    ! [A: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fr(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6683_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBTi,N: nat] :
      ( finite6580341952239402572_VEBTi(A)
     => finite5722435864697464412_VEBTi(collec1288327022334394266_VEBTi(aa_nat5293450157895600989i_bool(aTP_Lamm_fs(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6684_finite__lists__length__eq,axiom,
    ! [A: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ft(A),N))) ) ).

% finite_lists_length_eq
tff(fact_6685_powr__ge__pzero,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),Y3))) ).

% powr_ge_pzero
tff(fact_6686_powr__mono2,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),A2))) ) ) ) ).

% powr_mono2
tff(fact_6687_powr__less__mono,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B2))) ) ) ).

% powr_less_mono
tff(fact_6688_powr__less__cancel,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ) ).

% powr_less_cancel
tff(fact_6689_powr__mono,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B2))) ) ) ).

% powr_mono
tff(fact_6690_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X2: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y4: int] :
            ( pp(aa_set_int_bool(member_int(Y4),set_int2(Xs)))
           => ( Y4 = X2 ) )
       => ( Xs = replicate_int(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_6691_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y4),set_VEBT_VEBT2(Xs)))
           => ( Y4 = X2 ) )
       => ( Xs = replicate_VEBT_VEBT(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_6692_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X2: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y4: real] :
            ( pp(aa_set_real_bool(member_real(Y4),set_real2(Xs)))
           => ( Y4 = X2 ) )
       => ( Xs = replicate_real(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_6693_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X2: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y4: bool] :
            ( member_bool(Y4,set_bool2(Xs))
           => ( pp(Y4)
            <=> pp(X2) ) )
       => ( Xs = replicate_bool(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_6694_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X2: nat] :
      ( ( size_size_list_nat(Xs) = N )
     => ( ! [Y4: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y4),set_nat2(Xs)))
           => ( Y4 = X2 ) )
       => ( Xs = replicate_nat(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_6695_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBTi,N: nat,X2: vEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = N )
     => ( ! [Y4: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(Y4,set_VEBT_VEBTi2(Xs))
           => ( Y4 = X2 ) )
       => ( Xs = replicate_VEBT_VEBTi(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_6696_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs)))
         => ( X4 = X2 ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_6697_replicate__length__same,axiom,
    ! [Xs: list_real,X2: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => ( X4 = X2 ) )
     => ( replicate_real(size_size_list_real(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_6698_replicate__length__same,axiom,
    ! [Xs: list_bool,X2: bool] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => ( pp(X4)
          <=> pp(X2) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_6699_replicate__length__same,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( X4 = X2 ) )
     => ( replicate_nat(size_size_list_nat(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_6700_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ! [X4: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(Xs))
         => ( X4 = X2 ) )
     => ( replicate_VEBT_VEBTi(size_s7982070591426661849_VEBTi(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_6701_infinite__Icc,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A2,B2))) ) ).

% infinite_Icc
tff(fact_6702_infinite__Icc,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A2,B2))) ) ).

% infinite_Icc
tff(fact_6703_infinite__Ico,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or66887138388493659n_real(A2,B2))) ) ).

% infinite_Ico
tff(fact_6704_infinite__Ico,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A2,B2))) ) ).

% infinite_Ico
tff(fact_6705_finite__lists__length__le,axiom,
    ! [A: set_Code_integer,N: nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat3680331675976215733r_bool(aTP_Lamm_fu(A),N))) ) ).

% finite_lists_length_le
tff(fact_6706_finite__lists__length__le,axiom,
    ! [A: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_fv(A),N))) ) ).

% finite_lists_length_le
tff(fact_6707_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_fw(A),N))) ) ).

% finite_lists_length_le
tff(fact_6708_finite__lists__length__le,axiom,
    ! [A: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_fx(A),N))) ) ).

% finite_lists_length_le
tff(fact_6709_finite__lists__length__le,axiom,
    ! [A: set_bool,N: nat] :
      ( finite_finite_bool(A)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fy(A),N))) ) ).

% finite_lists_length_le
tff(fact_6710_finite__lists__length__le,axiom,
    ! [A: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fz(A),N))) ) ).

% finite_lists_length_le
tff(fact_6711_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBTi,N: nat] :
      ( finite6580341952239402572_VEBTi(A)
     => finite5722435864697464412_VEBTi(collec1288327022334394266_VEBTi(aa_nat5293450157895600989i_bool(aTP_Lamm_ga(A),N))) ) ).

% finite_lists_length_le
tff(fact_6712_finite__lists__length__le,axiom,
    ! [A: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_gb(A),N))) ) ).

% finite_lists_length_le
tff(fact_6713_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_nat(aTP_Lamm_gc(K),Lst) = replicate_nat(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_6714_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_real(aTP_Lamm_gd(K),Lst) = replicate_real(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_6715_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_VEBT_VEBT] : map_VE8901447254227204932T_VEBT(aTP_Lamm_ge(K),Lst) = replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_6716_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] : map_real_VEBT_VEBT(aTP_Lamm_gf(K),Lst) = replicate_VEBT_VEBT(size_size_list_real(Lst),K) ).

% map_replicate_const
tff(fact_6717_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_bool] : map_bool_VEBT_VEBT(aTP_Lamm_gg(K),Lst) = replicate_VEBT_VEBT(size_size_list_bool(Lst),K) ).

% map_replicate_const
tff(fact_6718_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_gh(K),Lst) = replicate_VEBT_VEBT(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_6719_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_VEBT_VEBTi] : map_VE7998069337340375161T_VEBT(aTP_Lamm_gi(K),Lst) = replicate_VEBT_VEBT(size_s7982070591426661849_VEBTi(Lst),K) ).

% map_replicate_const
tff(fact_6720_powr__mono2_H,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2))) ) ) ) ).

% powr_mono2'
tff(fact_6721_powr__less__mono2,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),A2))) ) ) ) ).

% powr_less_mono2
tff(fact_6722_gr__one__powr,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),Y3))) ) ) ).

% gr_one_powr
tff(fact_6723_powr__inj,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),X2) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Y3) )
        <=> ( X2 = Y3 ) ) ) ) ).

% powr_inj
tff(fact_6724_ge__one__powr__ge__zero,axiom,
    ! [X2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_6725_powr__mono__both,axiom,
    ! [A2: real,B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),B2))) ) ) ) ) ).

% powr_mono_both
tff(fact_6726_powr__le1,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_6727_powr__divide,axiom,
    ! [X2: real,Y3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),A2)) ) ) ) ).

% powr_divide
tff(fact_6728_powr__mult,axiom,
    ! [X2: real,Y3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),A2)) ) ) ) ).

% powr_mult
tff(fact_6729_powr__add,axiom,
    ! [X2: real,A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B2)) ).

% powr_add
tff(fact_6730_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z22)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z22)) ).

% powr_diff
tff(fact_6731_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_du,M)))) ) ).

% finite_divisors_nat
tff(fact_6732_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N6),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N6)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_6733_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N6),set_or1269000886237332187st_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N6)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_6734_set__replicate__Suc,axiom,
    ! [N: nat,X2: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X2)) = insert_real(X2,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_6735_set__replicate__Suc,axiom,
    ! [N: nat,X2: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X2)) = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_6736_set__replicate__Suc,axiom,
    ! [N: nat,X2: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X2)) = insert_nat(X2,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_6737_set__replicate__Suc,axiom,
    ! [N: nat,X2: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X2)) = insert_int(X2,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_6738_set__replicate__conv__if,axiom,
    ! [N: nat,X2: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X2)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X2)) = insert_real(X2,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_6739_set__replicate__conv__if,axiom,
    ! [N: nat,X2: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X2)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X2)) = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_6740_set__replicate__conv__if,axiom,
    ! [N: nat,X2: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X2)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X2)) = insert_nat(X2,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6741_set__replicate__conv__if,axiom,
    ! [N: nat,X2: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X2)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X2)) = insert_int(X2,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_6742_powr__less__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)),X2))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2))) ) ) ) ).

% powr_less_iff
tff(fact_6743_less__powr__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)),Y3)) ) ) ) ).

% less_powr_iff
tff(fact_6744_log__less__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)),Y3))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3))) ) ) ) ).

% log_less_iff
tff(fact_6745_less__log__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)),X2)) ) ) ) ).

% less_log_iff
tff(fact_6746_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_gj,N)))) ) ).

% finite_roots_unity
tff(fact_6747_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_gk,N)))) ) ).

% finite_roots_unity
tff(fact_6748_powr__mult__base,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y3)) ) ) ).

% powr_mult_base
tff(fact_6749_powr__le__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)),X2))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2))) ) ) ) ).

% powr_le_iff
tff(fact_6750_le__powr__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)),Y3)) ) ) ) ).

% le_powr_iff
tff(fact_6751_log__le__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)),Y3))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3))) ) ) ) ).

% log_le_iff
tff(fact_6752_le__log__iff,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)),X2)) ) ) ) ).

% le_log_iff
tff(fact_6753_log__add__eq__powr,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( ( B2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_6754_add__log__eq__powr,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( ( B2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)),X2)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_6755_minus__log__eq__powr,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( ( B2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y3)),X2)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_6756_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_6757_finite__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,B))))
    <=> pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A,B))) ) ).

% finite_Diff_insert
tff(fact_6758_finite__Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A,insert_int(A2,B))))
    <=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A,B))) ) ).

% finite_Diff_insert
tff(fact_6759_finite__Diff__insert,axiom,
    ! [A: set_Code_integer,A2: code_integer,B: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,insert_Code_integer(A2,B))))
    <=> pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B))) ) ).

% finite_Diff_insert
tff(fact_6760_finite__Diff__insert,axiom,
    ! [A: set_complex,A2: complex,B: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A,insert_complex(A2,B))))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A,B))) ) ).

% finite_Diff_insert
tff(fact_6761_finite__Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A,insert_nat(A2,B))))
    <=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A,B))) ) ).

% finite_Diff_insert
tff(fact_6762_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gl,K)))) ).

% finite_Collect_le_nat
tff(fact_6763_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,K)))) ).

% finite_Collect_less_nat
tff(fact_6764_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_gn(A))) ) ).

% finite_Collect_subsets
tff(fact_6765_finite__Collect__subsets,axiom,
    ! [A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_go(A))) ) ).

% finite_Collect_subsets
tff(fact_6766_finite__Collect__subsets,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_gp(A))) ) ).

% finite_Collect_subsets
tff(fact_6767_finite__Collect__subsets,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_gq(A))) ) ).

% finite_Collect_subsets
tff(fact_6768_finite__induct__select,axiom,
    ! [S: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [T4: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(T4),S))
             => ( pp(aa_set8260762808118238015T_bool(P,T4))
               => ? [X6: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),minus_5127226145743854075T_VEBT(S,T4)))
                    & pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X6,T4))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_6769_finite__induct__select,axiom,
    ! [S: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [T4: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(ord_le1307284697595431911nteger(T4),S))
             => ( pp(aa_set1374477347934578310r_bool(P,T4))
               => ? [X6: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),minus_2355218937544613996nteger(S,T4)))
                    & pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X6,T4))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_6770_finite__induct__select,axiom,
    ! [S: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [T4: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T4),S))
             => ( pp(aa_set_complex_bool(P,T4))
               => ? [X6: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X6),minus_811609699411566653omplex(S,T4)))
                    & pp(aa_set_complex_bool(P,insert_complex(X6,T4))) ) ) )
         => pp(aa_set_complex_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_6771_finite__induct__select,axiom,
    ! [S: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [T4: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T4),S))
             => ( pp(aa_set_int_bool(P,T4))
               => ? [X6: int] :
                    ( pp(aa_set_int_bool(member_int(X6),minus_minus_set_int(S,T4)))
                    & pp(aa_set_int_bool(P,insert_int(X6,T4))) ) ) )
         => pp(aa_set_int_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_6772_finite__induct__select,axiom,
    ! [S: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [T4: set_nat] :
              ( pp(aa_set_nat_bool(ord_less_set_nat(T4),S))
             => ( pp(aa_set_nat_bool(P,T4))
               => ? [X6: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X6),minus_minus_set_nat(S,T4)))
                    & pp(aa_set_nat_bool(P,insert_nat(X6,T4))) ) ) )
         => pp(aa_set_nat_bool(P,S)) ) ) ) ).

% finite_induct_select
tff(fact_6773_finite__Collect__disjI,axiom,
    ! [P: fun_Pr803113274115125473t_bool,Q: fun_Pr803113274115125473t_bool] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun8917503761599146496t_bool(aTP_Lamm_gr(P),Q)))
    <=> ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        & finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6774_finite__Collect__disjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_gs(P),Q))))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        & pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_6775_finite__Collect__disjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_gt(P),Q))))
    <=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        & pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_6776_finite__Collect__disjI,axiom,
    ! [P: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_gu(P),Q))))
    <=> ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
        & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_6777_finite__Collect__disjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_gv(P),Q))))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_6778_finite__Collect__conjI,axiom,
    ! [P: fun_Pr803113274115125473t_bool,Q: fun_Pr803113274115125473t_bool] :
      ( ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        | finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) )
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun8917503761599146496t_bool(aTP_Lamm_gw(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6779_finite__Collect__conjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        | pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) )
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_gx(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_6780_finite__Collect__conjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        | pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_gy(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_6781_finite__Collect__conjI,axiom,
    ! [P: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool] :
      ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
        | pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(Q))) )
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_gz(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_6782_finite__Collect__conjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ha(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_6783_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_hb(A2),B2)))) ).

% finite_interval_int1
tff(fact_6784_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_hc(A2),B2)))) ).

% finite_interval_int4
tff(fact_6785_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_or8404916559141939852nteger(L,U))) ).

% finite_atLeastLessThan_integer
tff(fact_6786_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_or189985376899183464nteger(L,U))) ).

% finite_atLeastAtMost_integer
tff(fact_6787_finite__Diff,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A,B))) ) ).

% finite_Diff
tff(fact_6788_finite__Diff,axiom,
    ! [A: set_Code_integer,B: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B))) ) ).

% finite_Diff
tff(fact_6789_finite__Diff,axiom,
    ! [A: set_complex,B: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A,B))) ) ).

% finite_Diff
tff(fact_6790_finite__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A,B))) ) ).

% finite_Diff
tff(fact_6791_finite__Diff2,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A,B)))
      <=> pp(aa_set_int_bool(finite_finite_int,A)) ) ) ).

% finite_Diff2
tff(fact_6792_finite__Diff2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B)))
      <=> pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A)) ) ) ).

% finite_Diff2
tff(fact_6793_finite__Diff2,axiom,
    ! [B: set_complex,A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A,B)))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A)) ) ) ).

% finite_Diff2
tff(fact_6794_finite__Diff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A,B)))
      <=> pp(aa_set_nat_bool(finite_finite_nat,A)) ) ) ).

% finite_Diff2
tff(fact_6795_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_hd(A2),B2)))) ).

% finite_interval_int2
tff(fact_6796_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_he(A2),B2)))) ).

% finite_interval_int3
tff(fact_6797_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U))) ).

% finite_atLeastZeroLessThan_integer
tff(fact_6798_finite__maxlen,axiom,
    ! [M8: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M8)
     => ? [N3: nat] :
        ! [X6: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X6,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X6)),N3)) ) ) ).

% finite_maxlen
tff(fact_6799_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( finite306553202115118035t_real(M8)
     => ? [N3: nat] :
        ! [X6: list_real] :
          ( member_list_real(X6,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_real(X6)),N3)) ) ) ).

% finite_maxlen
tff(fact_6800_finite__maxlen,axiom,
    ! [M8: set_list_bool] :
      ( finite7866827932528546246t_bool(M8)
     => ? [N3: nat] :
        ! [X6: list_bool] :
          ( member_list_bool(X6,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X6)),N3)) ) ) ).

% finite_maxlen
tff(fact_6801_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( finite8100373058378681591st_nat(M8)
     => ? [N3: nat] :
        ! [X6: list_nat] :
          ( member_list_nat(X6,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X6)),N3)) ) ) ).

% finite_maxlen
tff(fact_6802_finite__maxlen,axiom,
    ! [M8: set_list_VEBT_VEBTi] :
      ( finite5722435864697464412_VEBTi(M8)
     => ? [N3: nat] :
        ! [X6: list_VEBT_VEBTi] :
          ( member8117914210271334748_VEBTi(X6,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s7982070591426661849_VEBTi(X6)),N3)) ) ) ).

% finite_maxlen
tff(fact_6803_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_dv,I)))) ) ).

% finite_divisors_int
tff(fact_6804_pigeonhole__infinite__rel,axiom,
    ! [A: set_VEBT_VEBT,B: set_nat,R: fun_VE3763956218935105199t_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B))
                  & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(R,X4),Xa2)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_hf(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6805_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X4),Xa2)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_hg(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6806_pigeonhole__infinite__rel,axiom,
    ! [A: set_VEBT_VEBT,B: set_int,R: fun_VE3303673267288458579t_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B))
                  & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(R,X4),Xa2)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_hh(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6807_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X4),Xa2)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_hi(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6808_pigeonhole__infinite__rel,axiom,
    ! [A: set_VEBT_VEBT,B: set_Code_integer,R: fun_VE1123255499321568226r_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
             => ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),B))
                  & pp(aa_Code_integer_bool(aa_VEB6607917098650726363r_bool(R,X4),Xa2)) ) )
         => ? [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),B))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_Cod3526525079615536833T_bool(aa_fun7592865282689444384T_bool(aTP_Lamm_hj(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6809_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B: set_Code_integer,R: fun_re1120004649140997760r_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A))
             => ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),B))
                  & pp(aa_Code_integer_bool(aa_rea7928672538722535017r_bool(R,X4),Xa2)) ) )
         => ? [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_Cod181991726960451049l_bool(aa_fun8452463775060891964l_bool(aTP_Lamm_hk(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6810_pigeonhole__infinite__rel,axiom,
    ! [A: set_VEBT_VEBT,B: set_complex,R: fun_VE2128610149277289041x_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
             => ? [Xa2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa2),B))
                  & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(R,X4),Xa2)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_hl(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6811_pigeonhole__infinite__rel,axiom,
    ! [A: set_real,B: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A))
             => ? [Xa2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa2),B))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X4),Xa2)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_hm(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6812_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Xa2)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_hn(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6813_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X4),Xa2)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_ho(A),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6814_not__finite__existsD,axiom,
    ! [P: fun_Pr803113274115125473t_bool] :
      ( ~ finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ? [X_12: product_prod_int_int] : pp(aa_Pro4501881432599481050t_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_6815_not__finite__existsD,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ? [X_12: nat] : pp(aa_nat_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_6816_not__finite__existsD,axiom,
    ! [P: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_6817_not__finite__existsD,axiom,
    ! [P: fun_Co8822979462182099027r_bool] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
     => ? [X_12: code_integer] : pp(aa_Code_integer_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_6818_not__finite__existsD,axiom,
    ! [P: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ? [X_12: complex] : pp(aa_complex_bool(P,X_12)) ) ).

% not_finite_existsD
tff(fact_6819_finite__has__minimal2,axiom,
    ! [A: set_real,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(A2),A))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A2))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6820_finite__has__minimal2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
       => ? [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X4),A2))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6821_finite__has__minimal2,axiom,
    ! [A: set_set_int,A2: set_int] :
      ( finite6197958912794628473et_int(A)
     => ( member_set_int(A2,A)
       => ? [X4: set_int] :
            ( member_set_int(X4,A)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X4),A2))
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6822_finite__has__minimal2,axiom,
    ! [A: set_rat,A2: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A))
     => ( pp(aa_set_rat_bool(member_rat(A2),A))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),A2))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6823_finite__has__minimal2,axiom,
    ! [A: set_num,A2: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A))
     => ( pp(aa_set_num_bool(member_num(A2),A))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),A2))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6824_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(member_nat(A2),A))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),A2))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6825_finite__has__minimal2,axiom,
    ! [A: set_int,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(member_int(A2),A))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),A2))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_6826_finite__has__maximal2,axiom,
    ! [A: set_real,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(A2),A))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6827_finite__has__maximal2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
       => ? [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),X4))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6828_finite__has__maximal2,axiom,
    ! [A: set_set_int,A2: set_int] :
      ( finite6197958912794628473et_int(A)
     => ( member_set_int(A2,A)
       => ? [X4: set_int] :
            ( member_set_int(X4,A)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A2),X4))
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6829_finite__has__maximal2,axiom,
    ! [A: set_rat,A2: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A))
     => ( pp(aa_set_rat_bool(member_rat(A2),A))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),X4))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6830_finite__has__maximal2,axiom,
    ! [A: set_num,A2: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A))
     => ( pp(aa_set_num_bool(member_num(A2),A))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),X4))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6831_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(member_nat(A2),A))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X4))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6832_finite__has__maximal2,axiom,
    ! [A: set_int,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(member_int(A2),A))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X4))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_6833_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => pp(aa_set_nat_bool(finite_finite_nat,A)) ) ) ).

% finite_subset
tff(fact_6834_finite__subset,axiom,
    ! [A: set_Code_integer,B: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A)) ) ) ).

% finite_subset
tff(fact_6835_finite__subset,axiom,
    ! [A: set_complex,B: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A)) ) ) ).

% finite_subset
tff(fact_6836_finite__subset,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => pp(aa_set_int_bool(finite_finite_int,A)) ) ) ).

% finite_subset
tff(fact_6837_infinite__super,axiom,
    ! [S: set_nat,T5: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S),T5))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T5)) ) ) ).

% infinite_super
tff(fact_6838_infinite__super,axiom,
    ! [S: set_Code_integer,T5: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
     => ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5)) ) ) ).

% infinite_super
tff(fact_6839_infinite__super,axiom,
    ! [S: set_complex,T5: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T5)) ) ) ).

% infinite_super
tff(fact_6840_infinite__super,axiom,
    ! [S: set_int,T5: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S),T5))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S))
       => ~ pp(aa_set_int_bool(finite_finite_int,T5)) ) ) ).

% infinite_super
tff(fact_6841_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => pp(aa_set_nat_bool(finite_finite_nat,A)) ) ) ).

% rev_finite_subset
tff(fact_6842_rev__finite__subset,axiom,
    ! [B: set_Code_integer,A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A)) ) ) ).

% rev_finite_subset
tff(fact_6843_rev__finite__subset,axiom,
    ! [B: set_complex,A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A)) ) ) ).

% rev_finite_subset
tff(fact_6844_rev__finite__subset,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => pp(aa_set_int_bool(finite_finite_int,A)) ) ) ).

% rev_finite_subset
tff(fact_6845_Diff__infinite__finite,axiom,
    ! [T5: set_int,S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S))
       => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S,T5))) ) ) ).

% Diff_infinite_finite
tff(fact_6846_Diff__infinite__finite,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(S,T5))) ) ) ).

% Diff_infinite_finite
tff(fact_6847_Diff__infinite__finite,axiom,
    ! [T5: set_complex,S: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S,T5))) ) ) ).

% Diff_infinite_finite
tff(fact_6848_Diff__infinite__finite,axiom,
    ! [T5: set_nat,S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T5))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S,T5))) ) ) ).

% Diff_infinite_finite
tff(fact_6849_finite__has__maximal,axiom,
    ! [A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_6850_finite__has__maximal,axiom,
    ! [A: set_set_int] :
      ( finite6197958912794628473et_int(A)
     => ( ( A != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_6851_finite__has__maximal,axiom,
    ! [A: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A))
     => ( ( A != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_6852_finite__has__maximal,axiom,
    ! [A: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A))
     => ( ( A != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_6853_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_6854_finite__has__maximal,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_6855_finite__has__minimal,axiom,
    ! [A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_6856_finite__has__minimal,axiom,
    ! [A: set_set_int] :
      ( finite6197958912794628473et_int(A)
     => ( ( A != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_6857_finite__has__minimal,axiom,
    ! [A: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A))
     => ( ( A != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_6858_finite__has__minimal,axiom,
    ! [A: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A))
     => ( ( A != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_6859_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_6860_finite__has__minimal,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_6861_finite__subset__induct_H,axiom,
    ! [F3: set_VEBT_VEBT,A: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F3),A))
       => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F6))
               => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
                 => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F6),A))
                   => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),F6))
                     => ( pp(aa_set8260762808118238015T_bool(P,F6))
                       => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(A3,F6))) ) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6862_finite__subset__induct_H,axiom,
    ! [F3: set_real,A: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F6: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F6))
               => ( pp(aa_set_real_bool(member_real(A3),A))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F6),A))
                   => ( ~ pp(aa_set_real_bool(member_real(A3),F6))
                     => ( pp(aa_set_real_bool(P,F6))
                       => pp(aa_set_real_bool(P,insert_real(A3,F6))) ) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6863_finite__subset__induct_H,axiom,
    ! [F3: set_Code_integer,A: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F3))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(F3),A))
       => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F6))
               => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A))
                 => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(F6),A))
                   => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),F6))
                     => ( pp(aa_set1374477347934578310r_bool(P,F6))
                       => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(A3,F6))) ) ) ) ) )
           => pp(aa_set1374477347934578310r_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6864_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F6: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F6))
               => ( pp(aa_set_complex_bool(member_complex(A3),A))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F6),A))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A3),F6))
                     => ( pp(aa_set_complex_bool(P,F6))
                       => pp(aa_set_complex_bool(P,insert_complex(A3,F6))) ) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6865_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(F3),A))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F6: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F6))
               => ( pp(aa_set_nat_bool(member_nat(A3),A))
                 => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(F6),A))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A3),F6))
                     => ( pp(aa_set_nat_bool(P,F6))
                       => pp(aa_set_nat_bool(P,insert_nat(A3,F6))) ) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6866_finite__subset__induct_H,axiom,
    ! [F3: set_int,A: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F6: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F6))
               => ( pp(aa_set_int_bool(member_int(A3),A))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F6),A))
                   => ( ~ pp(aa_set_int_bool(member_int(A3),F6))
                     => ( pp(aa_set_int_bool(P,F6))
                       => pp(aa_set_int_bool(P,insert_int(A3,F6))) ) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6867_finite__subset__induct,axiom,
    ! [F3: set_VEBT_VEBT,A: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F3),A))
       => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F6))
               => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
                 => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),F6))
                   => ( pp(aa_set8260762808118238015T_bool(P,F6))
                     => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(A3,F6))) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6868_finite__subset__induct,axiom,
    ! [F3: set_real,A: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F6: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F6))
               => ( pp(aa_set_real_bool(member_real(A3),A))
                 => ( ~ pp(aa_set_real_bool(member_real(A3),F6))
                   => ( pp(aa_set_real_bool(P,F6))
                     => pp(aa_set_real_bool(P,insert_real(A3,F6))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6869_finite__subset__induct,axiom,
    ! [F3: set_Code_integer,A: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F3))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(F3),A))
       => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F6))
               => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A))
                 => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),F6))
                   => ( pp(aa_set1374477347934578310r_bool(P,F6))
                     => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(A3,F6))) ) ) ) )
           => pp(aa_set1374477347934578310r_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6870_finite__subset__induct,axiom,
    ! [F3: set_complex,A: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F6: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F6))
               => ( pp(aa_set_complex_bool(member_complex(A3),A))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A3),F6))
                   => ( pp(aa_set_complex_bool(P,F6))
                     => pp(aa_set_complex_bool(P,insert_complex(A3,F6))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6871_finite__subset__induct,axiom,
    ! [F3: set_nat,A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(F3),A))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F6: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F6))
               => ( pp(aa_set_nat_bool(member_nat(A3),A))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A3),F6))
                   => ( pp(aa_set_nat_bool(P,F6))
                     => pp(aa_set_nat_bool(P,insert_nat(A3,F6))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6872_finite__subset__induct,axiom,
    ! [F3: set_int,A: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F6: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F6))
               => ( pp(aa_set_int_bool(member_int(A3),A))
                 => ( ~ pp(aa_set_int_bool(member_int(A3),F6))
                   => ( pp(aa_set_int_bool(P,F6))
                     => pp(aa_set_int_bool(P,insert_int(A3,F6))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6873_infinite__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) ).

% infinite_remove
tff(fact_6874_infinite__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) ).

% infinite_remove
tff(fact_6875_infinite__remove,axiom,
    ! [S: set_complex,A2: complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) ).

% infinite_remove
tff(fact_6876_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
     => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S,insert_int(A2,bot_bot_set_int)))) ) ).

% infinite_remove
tff(fact_6877_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S,insert_nat(A2,bot_bot_set_nat)))) ) ).

% infinite_remove
tff(fact_6878_infinite__coinduct,axiom,
    ! [X: fun_se448528567104302406T_bool,A: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(X,A))
     => ( ! [A7: set_VEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(X,A7))
           => ? [X6: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),A7))
                & ( pp(aa_set8260762808118238015T_bool(X,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X6,bot_bo8194388402131092736T_VEBT))))
                  | ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X6,bot_bo8194388402131092736T_VEBT)))) ) ) )
       => ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A)) ) ) ).

% infinite_coinduct
tff(fact_6879_infinite__coinduct,axiom,
    ! [X: fun_se2712834123562403997r_bool,A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(X,A))
     => ( ! [A7: set_Code_integer] :
            ( pp(aa_set1374477347934578310r_bool(X,A7))
           => ? [X6: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A7))
                & ( pp(aa_set1374477347934578310r_bool(X,minus_2355218937544613996nteger(A7,insert_Code_integer(X6,bot_bo3990330152332043303nteger))))
                  | ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(A7,insert_Code_integer(X6,bot_bo3990330152332043303nteger)))) ) ) )
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A)) ) ) ).

% infinite_coinduct
tff(fact_6880_infinite__coinduct,axiom,
    ! [X: fun_set_complex_bool,A: set_complex] :
      ( pp(aa_set_complex_bool(X,A))
     => ( ! [A7: set_complex] :
            ( pp(aa_set_complex_bool(X,A7))
           => ? [X6: complex] :
                ( pp(aa_set_complex_bool(member_complex(X6),A7))
                & ( pp(aa_set_complex_bool(X,minus_811609699411566653omplex(A7,insert_complex(X6,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A7,insert_complex(X6,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A)) ) ) ).

% infinite_coinduct
tff(fact_6881_infinite__coinduct,axiom,
    ! [X: fun_set_int_bool,A: set_int] :
      ( pp(aa_set_int_bool(X,A))
     => ( ! [A7: set_int] :
            ( pp(aa_set_int_bool(X,A7))
           => ? [X6: int] :
                ( pp(aa_set_int_bool(member_int(X6),A7))
                & ( pp(aa_set_int_bool(X,minus_minus_set_int(A7,insert_int(X6,bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A7,insert_int(X6,bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A)) ) ) ).

% infinite_coinduct
tff(fact_6882_infinite__coinduct,axiom,
    ! [X: fun_set_nat_bool,A: set_nat] :
      ( pp(aa_set_nat_bool(X,A))
     => ( ! [A7: set_nat] :
            ( pp(aa_set_nat_bool(X,A7))
           => ? [X6: nat] :
                ( pp(aa_set_nat_bool(member_nat(X6),A7))
                & ( pp(aa_set_nat_bool(X,minus_minus_set_nat(A7,insert_nat(X6,bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A7,insert_nat(X6,bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A)) ) ) ).

% infinite_coinduct
tff(fact_6883_finite__empty__induct,axiom,
    ! [A: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set8260762808118238015T_bool(P,A))
       => ( ! [A3: vEBT_VEBT,A7: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A7))
             => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A7))
               => ( pp(aa_set8260762808118238015T_bool(P,A7))
                 => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(A3,bot_bo8194388402131092736T_VEBT)))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT)) ) ) ) ).

% finite_empty_induct
tff(fact_6884_finite__empty__induct,axiom,
    ! [A: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(P,A))
       => ( ! [A3: real,A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( pp(aa_set_real_bool(member_real(A3),A7))
               => ( pp(aa_set_real_bool(P,A7))
                 => pp(aa_set_real_bool(P,minus_minus_set_real(A7,insert_real(A3,bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_6885_finite__empty__induct,axiom,
    ! [A: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(P,A))
       => ( ! [A3: code_integer,A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A7))
               => ( pp(aa_set1374477347934578310r_bool(P,A7))
                 => pp(aa_set1374477347934578310r_bool(P,minus_2355218937544613996nteger(A7,insert_Code_integer(A3,bot_bo3990330152332043303nteger)))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger)) ) ) ) ).

% finite_empty_induct
tff(fact_6886_finite__empty__induct,axiom,
    ! [A: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(P,A))
       => ( ! [A3: complex,A7: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A7))
             => ( pp(aa_set_complex_bool(member_complex(A3),A7))
               => ( pp(aa_set_complex_bool(P,A7))
                 => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A7,insert_complex(A3,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_6887_finite__empty__induct,axiom,
    ! [A: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(P,A))
       => ( ! [A3: int,A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( pp(aa_set_int_bool(member_int(A3),A7))
               => ( pp(aa_set_int_bool(P,A7))
                 => pp(aa_set_int_bool(P,minus_minus_set_int(A7,insert_int(A3,bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_6888_finite__empty__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(P,A))
       => ( ! [A3: nat,A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( pp(aa_set_nat_bool(member_nat(A3),A7))
               => ( pp(aa_set_nat_bool(P,A7))
                 => pp(aa_set_nat_bool(P,minus_minus_set_nat(A7,insert_nat(A3,bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6889_finite__remove__induct,axiom,
    ! [B: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [A7: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A7))
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A7),B))
                 => ( ! [X6: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),A7))
                       => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X6,bot_bo8194388402131092736T_VEBT)))) )
                   => pp(aa_set8260762808118238015T_bool(P,A7)) ) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_6890_finite__remove__induct,axiom,
    ! [B: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( ( A7 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A7),B))
                 => ( ! [X6: real] :
                        ( pp(aa_set_real_bool(member_real(X6),A7))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A7,insert_real(X6,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A7)) ) ) ) )
         => pp(aa_set_real_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_6891_finite__remove__induct,axiom,
    ! [B: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A7),B))
                 => ( ! [X6: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A7))
                       => pp(aa_set1374477347934578310r_bool(P,minus_2355218937544613996nteger(A7,insert_Code_integer(X6,bot_bo3990330152332043303nteger)))) )
                   => pp(aa_set1374477347934578310r_bool(P,A7)) ) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_6892_finite__remove__induct,axiom,
    ! [B: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [A7: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A7))
             => ( ( A7 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A7),B))
                 => ( ! [X6: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X6),A7))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A7,insert_complex(X6,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A7)) ) ) ) )
         => pp(aa_set_complex_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_6893_finite__remove__induct,axiom,
    ! [B: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( ( A7 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A7),B))
                 => ( ! [X6: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X6),A7))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A7,insert_nat(X6,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A7)) ) ) ) )
         => pp(aa_set_nat_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_6894_finite__remove__induct,axiom,
    ! [B: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( ( A7 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A7),B))
                 => ( ! [X6: int] :
                        ( pp(aa_set_int_bool(member_int(X6),A7))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A7,insert_int(X6,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A7)) ) ) ) )
         => pp(aa_set_int_bool(P,B)) ) ) ) ).

% finite_remove_induct
tff(fact_6895_remove__induct,axiom,
    ! [P: fun_se448528567104302406T_bool,B: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
     => ( ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
         => pp(aa_set8260762808118238015T_bool(P,B)) )
       => ( ! [A7: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A7))
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A7),B))
                 => ( ! [X6: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X6),A7))
                       => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X6,bot_bo8194388402131092736T_VEBT)))) )
                   => pp(aa_set8260762808118238015T_bool(P,A7)) ) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_6896_remove__induct,axiom,
    ! [P: fun_set_real_bool,B: set_real] :
      ( pp(aa_set_real_bool(P,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B))
         => pp(aa_set_real_bool(P,B)) )
       => ( ! [A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( ( A7 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A7),B))
                 => ( ! [X6: real] :
                        ( pp(aa_set_real_bool(member_real(X6),A7))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A7,insert_real(X6,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A7)) ) ) ) )
         => pp(aa_set_real_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_6897_remove__induct,axiom,
    ! [P: fun_se2712834123562403997r_bool,B: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
     => ( ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
         => pp(aa_set1374477347934578310r_bool(P,B)) )
       => ( ! [A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A7),B))
                 => ( ! [X6: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A7))
                       => pp(aa_set1374477347934578310r_bool(P,minus_2355218937544613996nteger(A7,insert_Code_integer(X6,bot_bo3990330152332043303nteger)))) )
                   => pp(aa_set1374477347934578310r_bool(P,A7)) ) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_6898_remove__induct,axiom,
    ! [P: fun_set_complex_bool,B: set_complex] :
      ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
         => pp(aa_set_complex_bool(P,B)) )
       => ( ! [A7: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A7))
             => ( ( A7 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A7),B))
                 => ( ! [X6: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X6),A7))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A7,insert_complex(X6,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A7)) ) ) ) )
         => pp(aa_set_complex_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_6899_remove__induct,axiom,
    ! [P: fun_set_nat_bool,B: set_nat] :
      ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B))
         => pp(aa_set_nat_bool(P,B)) )
       => ( ! [A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( ( A7 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A7),B))
                 => ( ! [X6: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X6),A7))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A7,insert_nat(X6,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A7)) ) ) ) )
         => pp(aa_set_nat_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_6900_remove__induct,axiom,
    ! [P: fun_set_int_bool,B: set_int] :
      ( pp(aa_set_int_bool(P,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B))
         => pp(aa_set_int_bool(P,B)) )
       => ( ! [A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( ( A7 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A7),B))
                 => ( ! [X6: int] :
                        ( pp(aa_set_int_bool(member_int(X6),A7))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A7,insert_int(X6,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A7)) ) ) ) )
         => pp(aa_set_int_bool(P,B)) ) ) ) ).

% remove_induct
tff(fact_6901_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_hp(N),C2)))) ) ).

% finite_nth_roots
tff(fact_6902_set__encode__insert,axiom,
    ! [A: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A)) ) ) ) ).

% set_encode_insert
tff(fact_6903_diff__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,N6: fun_Pr5160692531401422620nt_nat] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_hq(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5604724294258851521t_bool(aTP_Lamm_hr(M8),N6))) ) ).

% diff_preserves_multiset
tff(fact_6904_diff__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,N6: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_hs(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_ht(M8),N6)))) ) ).

% diff_preserves_multiset
tff(fact_6905_diff__preserves__multiset,axiom,
    ! [M8: fun_int_nat,N6: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_hu(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_hv(M8),N6)))) ) ).

% diff_preserves_multiset
tff(fact_6906_diff__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,N6: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_hw(M8))))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun3540204951715082475r_bool(aTP_Lamm_hx(M8),N6)))) ) ).

% diff_preserves_multiset
tff(fact_6907_diff__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,N6: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_hy(M8))))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_hz(M8),N6)))) ) ).

% diff_preserves_multiset
tff(fact_6908_add__mset__in__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,A2: product_prod_int_int] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_hq(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(aTP_Lamm_ia(M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_6909_add__mset__in__multiset,axiom,
    ! [M8: fun_nat_nat,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_hs(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ib(M8),A2)))) ) ).

% add_mset_in_multiset
tff(fact_6910_add__mset__in__multiset,axiom,
    ! [M8: fun_int_nat,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_hu(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ic(M8),A2)))) ) ).

% add_mset_in_multiset
tff(fact_6911_add__mset__in__multiset,axiom,
    ! [M8: fun_Code_integer_nat,A2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_hw(M8))))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_id(M8),A2)))) ) ).

% add_mset_in_multiset
tff(fact_6912_add__mset__in__multiset,axiom,
    ! [M8: fun_complex_nat,A2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_hy(M8))))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ie(M8),A2)))) ) ).

% add_mset_in_multiset
tff(fact_6913_finite__linorder__max__induct,axiom,
    ! [A: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( ! [X6: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A7))
                   => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),B3)) )
               => ( pp(aa_set1374477347934578310r_bool(P,A7))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(B3,A7))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6914_finite__linorder__max__induct,axiom,
    ! [A: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( ! [X6: real] :
                    ( pp(aa_set_real_bool(member_real(X6),A7))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X6),B3)) )
               => ( pp(aa_set_real_bool(P,A7))
                 => pp(aa_set_real_bool(P,insert_real(B3,A7))) ) ) )
         => pp(aa_set_real_bool(P,A)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6915_finite__linorder__max__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A7: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A7))
             => ( ! [X6: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X6),A7))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X6),B3)) )
               => ( pp(aa_set_rat_bool(P,A7))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A7))) ) ) )
         => pp(aa_set_rat_bool(P,A)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6916_finite__linorder__max__induct,axiom,
    ! [A: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A7: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A7))
             => ( ! [X6: num] :
                    ( pp(aa_set_num_bool(member_num(X6),A7))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),B3)) )
               => ( pp(aa_set_num_bool(P,A7))
                 => pp(aa_set_num_bool(P,insert_num(B3,A7))) ) ) )
         => pp(aa_set_num_bool(P,A)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6917_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( ! [X6: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X6),A7))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),B3)) )
               => ( pp(aa_set_nat_bool(P,A7))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A7))) ) ) )
         => pp(aa_set_nat_bool(P,A)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6918_finite__linorder__max__induct,axiom,
    ! [A: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( ! [X6: int] :
                    ( pp(aa_set_int_bool(member_int(X6),A7))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),B3)) )
               => ( pp(aa_set_int_bool(P,A7))
                 => pp(aa_set_int_bool(P,insert_int(B3,A7))) ) ) )
         => pp(aa_set_int_bool(P,A)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6919_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B2),A2))),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,A2))) ).

% complex_mod_triangle_ineq2
tff(fact_6920_even__set__encode__iff,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) ) ) ).

% even_set_encode_iff
tff(fact_6921_infinite__growing,axiom,
    ! [X: set_Code_integer] :
      ( ( X != bot_bo3990330152332043303nteger )
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),X))
           => ? [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),X))
                & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),Xa2)) ) )
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,X)) ) ) ).

% infinite_growing
tff(fact_6922_infinite__growing,axiom,
    ! [X: set_real] :
      ( ( X != bot_bot_set_real )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),X))
           => ? [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),X))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Xa2)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X)) ) ) ).

% infinite_growing
tff(fact_6923_infinite__growing,axiom,
    ! [X: set_rat] :
      ( ( X != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),X))
           => ? [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),X))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Xa2)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X)) ) ) ).

% infinite_growing
tff(fact_6924_infinite__growing,axiom,
    ! [X: set_num] :
      ( ( X != bot_bot_set_num )
     => ( ! [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),X))
           => ? [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),X))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Xa2)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X)) ) ) ).

% infinite_growing
tff(fact_6925_infinite__growing,axiom,
    ! [X: set_nat] :
      ( ( X != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),X))
           => ? [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),X))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa2)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X)) ) ) ).

% infinite_growing
tff(fact_6926_infinite__growing,axiom,
    ! [X: set_int] :
      ( ( X != bot_bot_set_int )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),X))
           => ? [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),X))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Xa2)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X)) ) ) ).

% infinite_growing
tff(fact_6927_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
            & ~ ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),S))
                  & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6928_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( S != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S))
            & ~ ? [Xa2: real] :
                  ( pp(aa_set_real_bool(member_real(Xa2),S))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6929_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S))
     => ( ( S != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),S))
            & ~ ? [Xa2: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa2),S))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6930_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S))
     => ( ( S != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),S))
            & ~ ? [Xa2: num] :
                  ( pp(aa_set_num_bool(member_num(Xa2),S))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6931_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( S != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),S))
            & ~ ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),S))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6932_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( S != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),S))
            & ~ ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),S))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_6933_filter__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,P: fun_Pr803113274115125473t_bool] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_hq(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun8917503761599146496t_bool(aTP_Lamm_if(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_6934_filter__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_hs(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ig(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_6935_filter__preserves__multiset,axiom,
    ! [M8: fun_int_nat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_hu(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ih(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_6936_filter__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_hw(M8))))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_ii(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_6937_filter__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_hy(M8))))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ij(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_6938_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X4: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
             => ( ! [Y5: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y5),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y5)),aa_VEBT_VEBT_rat(F,X4))) )
               => ( pp(aa_set8260762808118238015T_bool(P,S5))
                 => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,S5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6939_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y5: real] :
                    ( pp(aa_set_real_bool(member_real(Y5),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y5)),aa_real_rat(F,X4))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,insert_real(X4,S5))) ) ) )
         => pp(aa_set_real_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6940_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: fun_se2712834123562403997r_bool,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [X4: code_integer,S5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S5))
             => ( ! [Y5: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y5),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,Y5)),aa_Code_integer_rat(F,X4))) )
               => ( pp(aa_set1374477347934578310r_bool(P,S5))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X4,S5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6941_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y5: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y5),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y5)),aa_complex_rat(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6942_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S5))
             => ( ! [Y5: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y5),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y5)),aa_nat_rat(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S5))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S5))) ) ) )
         => pp(aa_set_nat_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6943_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S5))
             => ( ! [Y5: int] :
                    ( pp(aa_set_int_bool(member_int(Y5),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y5)),aa_int_rat(F,X4))) )
               => ( pp(aa_set_int_bool(P,S5))
                 => pp(aa_set_int_bool(P,insert_int(X4,S5))) ) ) )
         => pp(aa_set_int_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6944_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_num] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X4: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
             => ( ! [Y5: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y5),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y5)),aa_VEBT_VEBT_num(F,X4))) )
               => ( pp(aa_set8260762808118238015T_bool(P,S5))
                 => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,S5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6945_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_bool,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y5: real] :
                    ( pp(aa_set_real_bool(member_real(Y5),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y5)),aa_real_num(F,X4))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,insert_real(X4,S5))) ) ) )
         => pp(aa_set_real_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6946_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: fun_se2712834123562403997r_bool,F: fun_Code_integer_num] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [X4: code_integer,S5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S5))
             => ( ! [Y5: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y5),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Code_integer_num(F,Y5)),aa_Code_integer_num(F,X4))) )
               => ( pp(aa_set1374477347934578310r_bool(P,S5))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X4,S5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6947_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y5: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y5),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y5)),aa_complex_num(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S)) ) ) ) ).

% finite_ranking_induct
tff(fact_6948_finite__linorder__min__induct,axiom,
    ! [A: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( ! [X6: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A7))
                   => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B3),X6)) )
               => ( pp(aa_set1374477347934578310r_bool(P,A7))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(B3,A7))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6949_finite__linorder__min__induct,axiom,
    ! [A: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( ! [X6: real] :
                    ( pp(aa_set_real_bool(member_real(X6),A7))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X6)) )
               => ( pp(aa_set_real_bool(P,A7))
                 => pp(aa_set_real_bool(P,insert_real(B3,A7))) ) ) )
         => pp(aa_set_real_bool(P,A)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6950_finite__linorder__min__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A7: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A7))
             => ( ! [X6: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X6),A7))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),X6)) )
               => ( pp(aa_set_rat_bool(P,A7))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A7))) ) ) )
         => pp(aa_set_rat_bool(P,A)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6951_finite__linorder__min__induct,axiom,
    ! [A: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A7: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A7))
             => ( ! [X6: num] :
                    ( pp(aa_set_num_bool(member_num(X6),A7))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),X6)) )
               => ( pp(aa_set_num_bool(P,A7))
                 => pp(aa_set_num_bool(P,insert_num(B3,A7))) ) ) )
         => pp(aa_set_num_bool(P,A)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6952_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( ! [X6: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X6),A7))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X6)) )
               => ( pp(aa_set_nat_bool(P,A7))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A7))) ) ) )
         => pp(aa_set_nat_bool(P,A)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6953_finite__linorder__min__induct,axiom,
    ! [A: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( ! [X6: int] :
                    ( pp(aa_set_int_bool(member_int(X6),A7))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X6)) )
               => ( pp(aa_set_int_bool(P,A7))
                 => pp(aa_set_int_bool(P,insert_int(B3,A7))) ) ) )
         => pp(aa_set_int_bool(P,A)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6954_freeze__rule,axiom,
    ! [A2: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3904069481286416050_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A2),Xs),array_8141364883501958055_VEBTi(A2),aa_lis3169797078229555565i_assn(aTP_Lamm_ik(A2),Xs)) ).

% freeze_rule
tff(fact_6955_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_assn,Y3: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun437439898845168857T_bool(aTP_Lamm_il(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun437439898845168857T_bool(aTP_Lamm_il(I6),Y3))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun437439898845168857T_bool(aa_fun2621268342929013301T_bool(aTP_Lamm_im(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6956_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_assn,Y3: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun159070158279343989l_bool(aTP_Lamm_in(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun159070158279343989l_bool(aTP_Lamm_in(I6),Y3))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun159070158279343989l_bool(aa_fun6385136150007740861l_bool(aTP_Lamm_io(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6957_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_assn,Y3: fun_nat_assn] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1272683333798124589t_bool(aTP_Lamm_ip(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1272683333798124589t_bool(aTP_Lamm_ip(I6),Y3))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1272683333798124589t_bool(aa_fun7103831392407232081t_bool(aTP_Lamm_iq(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6958_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_assn,Y3: fun_int_assn] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1414614411059211637t_bool(aTP_Lamm_ir(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1414614411059211637t_bool(aTP_Lamm_ir(I6),Y3))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1414614411059211637t_bool(aa_fun5523883950577104445t_bool(aTP_Lamm_is(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6959_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_Code_integer,X2: fun_Co4490034186337852034r_assn,Y3: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8795547071341169043r_bool(aTP_Lamm_it(I6),X2))))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8795547071341169043r_bool(aTP_Lamm_it(I6),Y3))))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8795547071341169043r_bool(aa_fun7758107644544034538r_bool(aTP_Lamm_iu(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6960_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: fun_complex_assn,Y3: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1925668587588942193x_bool(aTP_Lamm_iv(I6),X2))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1925668587588942193x_bool(aTP_Lamm_iv(I6),Y3))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1925668587588942193x_bool(aa_fun8862831692315842871x_bool(aTP_Lamm_iw(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6961_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_real,Y3: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_ix(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_ix(I6),Y3))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_iy(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6962_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_real,Y3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_iz(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_iz(I6),Y3))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ja(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6963_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_real,Y3: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_jb(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_jb(I6),Y3))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_jc(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6964_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_real,Y3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_jd(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_jd(I6),Y3))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_je(I6),X2),Y3)))) ) ) ).

% prod.finite_Collect_op
tff(fact_6965_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VE5569070535936029429omplex,Y3: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_jf(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_jf(I6),Y3))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_jg(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6966_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_complex,Y3: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_jh(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_jh(I6),Y3))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_ji(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6967_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_complex,Y3: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_jj(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_jj(I6),Y3))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_jk(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6968_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_complex,Y3: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_jl(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_jl(I6),Y3))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_jm(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6969_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Code_integer,X2: fun_Co1900716531650368264omplex,Y3: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_jn(I6),X2))))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_jn(I6),Y3))))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aa_fun6901856055265004766r_bool(aTP_Lamm_jo(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6970_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: fun_complex_complex,Y3: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_jp(I6),X2))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_jp(I6),Y3))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_jq(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6971_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_real,Y3: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_jr(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_jr(I6),Y3))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_js(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6972_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_real,Y3: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_jt(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_jt(I6),Y3))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ju(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6973_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_real,Y3: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_jv(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_jv(I6),Y3))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_jw(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6974_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_real,Y3: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_jx(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_jx(I6),Y3))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_jy(I6),X2),Y3)))) ) ) ).

% sum.finite_Collect_op
tff(fact_6975_sum__gp,axiom,
    ! [N: nat,M: nat,X2: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X2 = one_one_complex )
           => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X2 != one_one_complex )
           => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_nat(suc,N)))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)) ) ) ) ) ) ).

% sum_gp
tff(fact_6976_sum__gp,axiom,
    ! [N: nat,M: nat,X2: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X2 = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X2 != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),M)),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ) ) ).

% sum_gp
tff(fact_6977_sum__gp,axiom,
    ! [N: nat,M: nat,X2: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X2 = one_one_real )
           => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X2 != one_one_real )
           => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ) ) ).

% sum_gp
tff(fact_6978_sum_Oneutral__const,axiom,
    ! [A: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bj),A) = zero_zero_nat ).

% sum.neutral_const
tff(fact_6979_sum_Oneutral__const,axiom,
    ! [A: set_complex] : groups7754918857620584856omplex(aTP_Lamm_bg,A) = zero_zero_complex ).

% sum.neutral_const
tff(fact_6980_sum_Oneutral__const,axiom,
    ! [A: set_nat] : groups6591440286371151544t_real(aTP_Lamm_jz,A) = zero_zero_real ).

% sum.neutral_const
tff(fact_6981_sum_Oneutral__const,axiom,
    ! [A: set_int] : groups4538972089207619220nt_int(aTP_Lamm_bk,A) = zero_zero_int ).

% sum.neutral_const
tff(fact_6982_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A)) = groups7754918857620584856omplex(aTP_Lamm_ka(F),A) ).

% of_nat_sum
tff(fact_6983_of__nat__sum,axiom,
    ! [F: fun_int_nat,A: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A)) = groups4538972089207619220nt_int(aTP_Lamm_kb(F),A) ).

% of_nat_sum
tff(fact_6984_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups3539618377306564664at_int(aTP_Lamm_kc(F),A) ).

% of_nat_sum
tff(fact_6985_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kd(F)),A) ).

% of_nat_sum
tff(fact_6986_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups6591440286371151544t_real(aTP_Lamm_ke(F),A) ).

% of_nat_sum
tff(fact_6987_of__int__sum,axiom,
    ! [F: fun_complex_int,A: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A)) = groups7754918857620584856omplex(aTP_Lamm_kf(F),A) ).

% of_int_sum
tff(fact_6988_of__int__sum,axiom,
    ! [F: fun_nat_int,A: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A)) = groups6591440286371151544t_real(aTP_Lamm_kg(F),A) ).

% of_int_sum
tff(fact_6989_of__int__sum,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A)) = groups8778361861064173332t_real(aTP_Lamm_kh(F),A) ).

% of_int_sum
tff(fact_6990_of__int__sum,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A)) = groups3906332499630173760nt_rat(aTP_Lamm_ki(F),A) ).

% of_int_sum
tff(fact_6991_of__int__sum,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A)) = groups4538972089207619220nt_int(aTP_Lamm_kj(F),A) ).

% of_int_sum
tff(fact_6992_sum_Odelta_H,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_kk(A2),B2),S) = aa_VEBT_VEBT_complex(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_kk(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6993_sum_Odelta_H,axiom,
    ! [S: set_real,A2: real,B2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_kl(A2),B2),S) = aa_real_complex(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_kl(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6994_sum_Odelta_H,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_km(A2),B2),S) = aa_nat_complex(B2,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_km(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6995_sum_Odelta_H,axiom,
    ! [S: set_int,A2: int,B2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_kn(A2),B2),S) = aa_int_complex(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_kn(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6996_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ko(A2),B2),S) = aa_Cod5621350732091595593omplex(B2,A2) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ko(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_6997_sum_Odelta_H,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_kp(A2),B2),S) = aa_VEBT_VEBT_real(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_kp(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6998_sum_Odelta_H,axiom,
    ! [S: set_real,A2: real,B2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kq(A2),B2),S) = aa_real_real(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kq(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6999_sum_Odelta_H,axiom,
    ! [S: set_int,A2: int,B2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_kr(A2),B2),S) = aa_int_real(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_kr(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7000_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_ks(A2),B2),S) = aa_Code_integer_real(B2,A2) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_ks(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7001_sum_Odelta_H,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_kt(A2),B2),S) = aa_complex_real(B2,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_kt(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7002_sum_Odelta,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ku(A2),B2),S) = aa_VEBT_VEBT_complex(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ku(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_7003_sum_Odelta,axiom,
    ! [S: set_real,A2: real,B2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_kv(A2),B2),S) = aa_real_complex(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_kv(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_7004_sum_Odelta,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kw(A2),B2),S) = aa_nat_complex(B2,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kw(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_7005_sum_Odelta,axiom,
    ! [S: set_int,A2: int,B2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_kx(A2),B2),S) = aa_int_complex(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_kx(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_7006_sum_Odelta,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ky(A2),B2),S) = aa_Cod5621350732091595593omplex(B2,A2) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ky(A2),B2),S) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_7007_sum_Odelta,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_kz(A2),B2),S) = aa_VEBT_VEBT_real(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_kz(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7008_sum_Odelta,axiom,
    ! [S: set_real,A2: real,B2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_la(A2),B2),S) = aa_real_real(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_la(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7009_sum_Odelta,axiom,
    ! [S: set_int,A2: int,B2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_lb(A2),B2),S) = aa_int_real(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_lb(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7010_sum_Odelta,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_lc(A2),B2),S) = aa_Code_integer_real(B2,A2) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_lc(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7011_sum_Odelta,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ld(A2),B2),S) = aa_complex_real(B2,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ld(A2),B2),S) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7012_sum_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7013_sum_Oinsert,axiom,
    ! [A: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A))
       => ( groups8097168146408367636l_real(G,insert_real(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7014_sum_Oinsert,axiom,
    ! [A: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A))
       => ( groups8778361861064173332t_real(G,insert_int(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X2)),groups8778361861064173332t_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7015_sum_Oinsert,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups1270011288395367621r_real(G,insert_Code_integer(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,X2)),groups1270011288395367621r_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7016_sum_Oinsert,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ~ pp(aa_set_complex_bool(member_complex(X2),A))
       => ( groups5808333547571424918x_real(G,insert_complex(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,A)) ) ) ) ).

% sum.insert
tff(fact_7017_sum_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7018_sum_Oinsert,axiom,
    ! [A: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A))
       => ( groups1300246762558778688al_rat(G,insert_real(X2,A)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7019_sum_Oinsert,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
       => ( groups2906978787729119204at_rat(G,insert_nat(X2,A)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,X2)),groups2906978787729119204at_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7020_sum_Oinsert,axiom,
    ! [A: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A))
       => ( groups3906332499630173760nt_rat(G,insert_int(X2,A)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X2)),groups3906332499630173760nt_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7021_sum_Oinsert,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups6602215022474089585er_rat(G,insert_Code_integer(X2,A)) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,X2)),groups6602215022474089585er_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_7022_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups2073611262835488442omplex(G,set_or4665077453230672383an_nat(M,N))),aa_nat_complex(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_7023_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_7024_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,N))),aa_nat_int(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_7025_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,N))),aa_nat_nat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_7026_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,N))),aa_nat_real(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_7027_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_7028_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_7029_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_7030_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_7031_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_7032_sum__zero__power,axiom,
    ! [A: set_nat,C2: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_le(C2),A) = aa_nat_complex(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_le(C2),A) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_7033_sum__zero__power,axiom,
    ! [A: set_nat,C2: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_lf(C2),A) = aa_nat_rat(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_lf(C2),A) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_7034_sum__zero__power,axiom,
    ! [A: set_nat,C2: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups6591440286371151544t_real(aTP_Lamm_lg(C2),A) = aa_nat_real(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups6591440286371151544t_real(aTP_Lamm_lg(C2),A) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_7035_sum__zero__power_H,axiom,
    ! [A: set_nat,C2: fun_nat_complex,D2: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_lh(C2),D2),A) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(C2,zero_zero_nat)),aa_nat_complex(D2,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_lh(C2),D2),A) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_7036_sum__zero__power_H,axiom,
    ! [A: set_nat,C2: fun_nat_rat,D2: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_li(C2),D2),A) = aa_rat_rat(divide_divide_rat(aa_nat_rat(C2,zero_zero_nat)),aa_nat_rat(D2,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_li(C2),D2),A) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_7037_sum__zero__power_H,axiom,
    ! [A: set_nat,C2: fun_nat_real,D2: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lj(C2),D2),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C2,zero_zero_nat)),aa_nat_real(D2,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A)) )
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lj(C2),D2),A) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_7038_sum_Oswap__restrict,axiom,
    ! [A: set_VEBT_VEBT,B: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups771621172384141258BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_ll(B),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_ln(A),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7039_sum_Oswap__restrict,axiom,
    ! [A: set_real,B: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_lp(B),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_lr(A),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7040_sum_Oswap__restrict,axiom,
    ! [A: set_int,B: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_lt(B),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_lw(A),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7041_sum_Oswap__restrict,axiom,
    ! [A: set_Code_integer,B: set_nat,G: fun_Co3312552843764780407at_nat,R: fun_Co8657364630979331932t_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups7237345082560585321er_nat(aa_fun2949368542467568330er_nat(aa_fun8366171557965857828er_nat(aTP_Lamm_ly(B),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7549014261834767933at_nat(aa_fun5721305349739569687at_nat(aTP_Lamm_mb(A),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7042_sum_Oswap__restrict,axiom,
    ! [A: set_complex,B: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_md(B),G),R),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_mg(A),G),R)),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7043_sum_Oswap__restrict,axiom,
    ! [A: set_VEBT_VEBT,B: set_complex,G: fun_VE2006399705683203104omplex,R: fun_VE2128610149277289041x_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups1794756597179926696omplex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_mi(B),G),R),A) = groups7754918857620584856omplex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_mk(A),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7044_sum_Oswap__restrict,axiom,
    ! [A: set_real,B: set_complex,G: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_mm(B),G),R),A) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_mo(A),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7045_sum_Oswap__restrict,axiom,
    ! [A: set_nat,B: set_complex,G: fun_na1591161823130732966omplex,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups2073611262835488442omplex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_mq(B),G),R),A) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_mt(A),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7046_sum_Oswap__restrict,axiom,
    ! [A: set_int,B: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5750378413879446255x_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups3049146728041665814omplex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_mv(B),G),R),A) = groups7754918857620584856omplex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_my(A),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7047_sum_Oswap__restrict,axiom,
    ! [A: set_Code_integer,B: set_complex,G: fun_Co8905303033605938227omplex,R: fun_Co7124838579184476414x_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
       => ( groups8024822376189712711omplex(aa_fun7983975303565649482omplex(aa_fun843835629537646824omplex(aTP_Lamm_na(B),G),R),A) = groups7754918857620584856omplex(aa_fun8169706760919072411omplex(aa_fun2334956106851614393omplex(aTP_Lamm_nd(A),G),R),B) ) ) ) ).

% sum.swap_restrict
tff(fact_7048_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ne(F),G),A) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A)),groups7754918857620584856omplex(G,A)) ).

% sum_subtractf
tff(fact_7049_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nf(F),G),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A)),groups6591440286371151544t_real(G,A)) ).

% sum_subtractf
tff(fact_7050_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ng(F),G),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A)),groups4538972089207619220nt_int(G,A)) ).

% sum_subtractf
tff(fact_7051_sum__norm__le,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,G: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_VEBT_VEBT_complex(F,X4))),aa_VEBT_VEBT_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups1794756597179926696omplex(F,S))),groups2240296850493347238T_real(G,S))) ) ).

% sum_norm_le
tff(fact_7052_sum__norm__le,axiom,
    ! [S: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X4))),aa_real_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S))),groups8097168146408367636l_real(G,S))) ) ).

% sum_norm_le
tff(fact_7053_sum__norm__le,axiom,
    ! [S: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X4))),aa_int_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S))),groups8778361861064173332t_real(G,S))) ) ).

% sum_norm_le
tff(fact_7054_sum__norm__le,axiom,
    ! [S: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X4))),aa_nat_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,S))),groups6591440286371151544t_real(G,S))) ) ).

% sum_norm_le
tff(fact_7055_sum__norm__le,axiom,
    ! [S: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X4))),aa_complex_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S))),groups5808333547571424918x_real(G,S))) ) ).

% sum_norm_le
tff(fact_7056_sum__norm__le,axiom,
    ! [S: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X4))),aa_nat_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,S))),groups6591440286371151544t_real(G,S))) ) ).

% sum_norm_le
tff(fact_7057_sum__mono,axiom,
    ! [K6: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K6))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K6)),groups2906978787729119204at_rat(G,K6))) ) ).

% sum_mono
tff(fact_7058_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),K6))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,K6)),groups136491112297645522BT_rat(G,K6))) ) ).

% sum_mono
tff(fact_7059_sum__mono,axiom,
    ! [K6: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K6))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K6)),groups1300246762558778688al_rat(G,K6))) ) ).

% sum_mono
tff(fact_7060_sum__mono,axiom,
    ! [K6: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K6))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K6)),groups3906332499630173760nt_rat(G,K6))) ) ).

% sum_mono
tff(fact_7061_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),K6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,K6)),groups771621172384141258BT_nat(G,K6))) ) ).

% sum_mono
tff(fact_7062_sum__mono,axiom,
    ! [K6: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K6)),groups1935376822645274424al_nat(G,K6))) ) ).

% sum_mono
tff(fact_7063_sum__mono,axiom,
    ! [K6: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K6)),groups4541462559716669496nt_nat(G,K6))) ) ).

% sum_mono
tff(fact_7064_sum__mono,axiom,
    ! [K6: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K6)),groups3539618377306564664at_int(G,K6))) ) ).

% sum_mono
tff(fact_7065_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),K6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_VEBT_VEBT_int(F,I2)),aa_VEBT_VEBT_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups769130701875090982BT_int(F,K6)),groups769130701875090982BT_int(G,K6))) ) ).

% sum_mono
tff(fact_7066_sum__mono,axiom,
    ! [K6: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K6)),groups1932886352136224148al_int(G,K6))) ) ).

% sum_mono
tff(fact_7067_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nh(R2),F)),A) ).

% sum_distrib_left
tff(fact_7068_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),groups7754918857620584856omplex(F,A)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ni(R2),F),A) ).

% sum_distrib_left
tff(fact_7069_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),groups6591440286371151544t_real(F,A)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nj(R2),F),A) ).

% sum_distrib_left
tff(fact_7070_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_nk(R2),F),A) ).

% sum_distrib_left
tff(fact_7071_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nl(F),R2)),A) ).

% sum_distrib_right
tff(fact_7072_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_nm(F),R2),A) ).

% sum_distrib_right
tff(fact_7073_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nn(F),R2),A) ).

% sum_distrib_right
tff(fact_7074_sum__distrib__right,axiom,
    ! [F: fun_int_int,A: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_no(F),R2),A) ).

% sum_distrib_right
tff(fact_7075_sum__product,axiom,
    ! [F: fun_nat_nat,A: set_nat,G: fun_nat_nat,B: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_nq(F),G),B)),A) ).

% sum_product
tff(fact_7076_sum__product,axiom,
    ! [F: fun_complex_complex,A: set_complex,G: fun_complex_complex,B: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A)),groups7754918857620584856omplex(G,B)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ns(F),G),B),A) ).

% sum_product
tff(fact_7077_sum__product,axiom,
    ! [F: fun_nat_real,A: set_nat,G: fun_nat_real,B: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A)),groups6591440286371151544t_real(G,B)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_nu(F),G),B),A) ).

% sum_product
tff(fact_7078_sum__product,axiom,
    ! [F: fun_int_int,A: set_int,G: fun_int_int,B: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A)),groups4538972089207619220nt_int(G,B)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_nw(F),G),B),A) ).

% sum_product
tff(fact_7079_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B: set_nat,A: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_nx(G),B)),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_nz(G),A)),B) ).

% sum.swap
tff(fact_7080_sum_Oswap,axiom,
    ! [G: fun_co7673397065115134084omplex,B: set_complex,A: set_complex] : groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_oa(G),B),A) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_oc(G),A),B) ).

% sum.swap
tff(fact_7081_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_real,B: set_nat,A: set_nat] : groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_od(G),B),A) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_of(G),A),B) ).

% sum.swap
tff(fact_7082_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B: set_int,A: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_og(G),B),A) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_oi(G),A),B) ).

% sum.swap
tff(fact_7083_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups7754918857620584856omplex(F,A)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_oj(F),R2),A) ).

% sum_divide_distrib
tff(fact_7084_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ok(F),R2),A) ).

% sum_divide_distrib
tff(fact_7085_norm__sum,axiom,
    ! [F: fun_nat_complex,A: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,A))),groups6591440286371151544t_real(aTP_Lamm_ol(F),A))) ).

% norm_sum
tff(fact_7086_norm__sum,axiom,
    ! [F: fun_complex_complex,A: set_complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A))),groups5808333547571424918x_real(aTP_Lamm_om(F),A))) ).

% norm_sum
tff(fact_7087_norm__sum,axiom,
    ! [F: fun_nat_real,A: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,A))),groups6591440286371151544t_real(aTP_Lamm_on(F),A))) ).

% norm_sum
tff(fact_7088_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oo(G),H)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A)) ).

% sum.distrib
tff(fact_7089_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_op(G),H),A) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups7754918857620584856omplex(G,A)),groups7754918857620584856omplex(H,A)) ).

% sum.distrib
tff(fact_7090_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_oq(G),H),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A)),groups6591440286371151544t_real(H,A)) ).

% sum.distrib
tff(fact_7091_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_or(G),H),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A)),groups4538972089207619220nt_int(H,A)) ).

% sum.distrib
tff(fact_7092_sum__nonpos,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_7093_sum__nonpos,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_7094_sum__nonpos,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_7095_sum__nonpos,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7096_sum__nonpos,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7097_sum__nonpos,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7098_sum__nonpos,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7099_sum__nonpos,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,A)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_7100_sum__nonpos,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_7101_sum__nonpos,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_7102_sum__nonneg,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(F,A))) ) ).

% sum_nonneg
tff(fact_7103_sum__nonneg,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A))) ) ).

% sum_nonneg
tff(fact_7104_sum__nonneg,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A))) ) ).

% sum_nonneg
tff(fact_7105_sum__nonneg,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A))) ) ).

% sum_nonneg
tff(fact_7106_sum__nonneg,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A))) ) ).

% sum_nonneg
tff(fact_7107_sum__nonneg,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A))) ) ).

% sum_nonneg
tff(fact_7108_sum__nonneg,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A))) ) ).

% sum_nonneg
tff(fact_7109_sum__nonneg,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups771621172384141258BT_nat(F,A))) ) ).

% sum_nonneg
tff(fact_7110_sum__nonneg,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A))) ) ).

% sum_nonneg
tff(fact_7111_sum__nonneg,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A))) ) ).

% sum_nonneg
tff(fact_7112_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_rat,I6: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(F,I6) = groups136491112297645522BT_rat(G,I6) )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2))) )
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I6))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
           => ( aa_VEBT_VEBT_rat(F,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7113_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I6) = groups1300246762558778688al_rat(G,I6) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
       => ( pp(aa_set_real_bool(member_real(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7114_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I6) = groups2906978787729119204at_rat(G,I6) )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I6))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7115_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I6) = groups3906332499630173760nt_rat(G,I6) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7116_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_rat,I6: set_Code_integer,G: fun_Code_integer_rat,I: code_integer] :
      ( ( groups6602215022474089585er_rat(F,I6) = groups6602215022474089585er_rat(G,I6) )
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,I2)),aa_Code_integer_rat(G,I2))) )
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I6))
         => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
           => ( aa_Code_integer_rat(F,I) = aa_Code_integer_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7117_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I6) = groups5058264527183730370ex_rat(G,I6) )
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7118_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_nat,I6: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(F,I6) = groups771621172384141258BT_nat(G,I6) )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2))) )
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I6))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
           => ( aa_VEBT_VEBT_nat(F,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7119_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
       => ( pp(aa_set_real_bool(member_real(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7120_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7121_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_nat,I6: set_Code_integer,G: fun_Code_integer_nat,I: code_integer] :
      ( ( groups7237345082560585321er_nat(F,I6) = groups7237345082560585321er_nat(G,I6) )
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Code_integer_nat(F,I2)),aa_Code_integer_nat(G,I2))) )
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I6))
         => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
           => ( aa_Code_integer_nat(F,I) = aa_Code_integer_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7122_sum__cong__Suc,axiom,
    ! [A: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A) ) ) ) ).

% sum_cong_Suc
tff(fact_7123_sum__cong__Suc,axiom,
    ! [A: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A))
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( groups6591440286371151544t_real(F,A) = groups6591440286371151544t_real(G,A) ) ) ) ).

% sum_cong_Suc
tff(fact_7124_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A2: nat,A: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_os(F),A2)),A)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),A2) ).

% mod_sum_eq
tff(fact_7125_mod__sum__eq,axiom,
    ! [F: fun_int_int,A2: int,A: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ot(F),A2),A)),A2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A)),A2) ).

% mod_sum_eq
tff(fact_7126_sum_Ointer__filter,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups1794756597179926696omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_au(A),P))) = groups1794756597179926696omplex(aa_fun6792494735270171281omplex(aTP_Lamm_ou(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7127_sum_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_av(A),P))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_ov(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7128_sum_Ointer__filter,axiom,
    ! [A: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(A),P))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_ow(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7129_sum_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_az(A),P))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_ox(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7130_sum_Ointer__filter,axiom,
    ! [A: set_Code_integer,G: fun_Co1900716531650368264omplex,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups8024822376189712711omplex(G,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_oy(A),P))) = groups8024822376189712711omplex(aa_fun5444226037728322975omplex(aTP_Lamm_oz(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7131_sum_Ointer__filter,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_au(A),P))) = groups2240296850493347238T_real(aa_fun2072160908118572431T_real(aTP_Lamm_pa(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7132_sum_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_av(A),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_pb(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7133_sum_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_az(A),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_pc(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7134_sum_Ointer__filter,axiom,
    ! [A: set_Code_integer,G: fun_Co3960896333048504454r_real,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups1270011288395367621r_real(G,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_oy(A),P))) = groups1270011288395367621r_real(aa_fun1813952354503624093r_real(aTP_Lamm_pd(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7135_sum_Ointer__filter,axiom,
    ! [A: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ax(A),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_pe(G),P),A) ) ) ).

% sum.inter_filter
tff(fact_7136_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),T))
                    & ( aa_int_int(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7137_sum__le__included,axiom,
    ! [S2: set_int,T: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Code_integer_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ? [Xa2: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),T))
                    & ( aa_Code_integer_int(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_Code_integer_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups1270011288395367621r_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7138_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ? [Xa2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa2),T))
                    & ( aa_complex_int(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_complex_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7139_sum__le__included,axiom,
    ! [S2: set_Code_integer,T: set_int,G: fun_int_real,I: fun_int_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S2))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),T))
                    & ( aa_int_Code_integer(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Code_integer_real(F,X4)),aa_int_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1270011288395367621r_real(F,S2)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7140_sum__le__included,axiom,
    ! [S2: set_Code_integer,T: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co8499238673658658103nteger,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X4))) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S2))
               => ? [Xa2: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),T))
                    & ( aa_Cod1783974652533396728nteger(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Code_integer_real(F,X4)),aa_Code_integer_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1270011288395367621r_real(F,S2)),groups1270011288395367621r_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7141_sum__le__included,axiom,
    ! [S2: set_Code_integer,T: set_complex,G: fun_complex_real,I: fun_co8153928105989009288nteger,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S2))
               => ? [Xa2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa2),T))
                    & ( aa_com4217951221296830281nteger(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Code_integer_real(F,X4)),aa_complex_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1270011288395367621r_real(F,S2)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7142_sum__le__included,axiom,
    ! [S2: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),T))
                    & ( aa_int_complex(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_int_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7143_sum__le__included,axiom,
    ! [S2: set_complex,T: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co1900716531650368264omplex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ? [Xa2: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),T))
                    & ( aa_Cod5621350732091595593omplex(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_Code_integer_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups1270011288395367621r_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7144_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ? [Xa2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa2),T))
                    & ( aa_complex_complex(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7145_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => ? [Xa2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa2),T))
                    & ( aa_nat_nat(I,Xa2) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa2))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7146_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
       => ( ( groups2240296850493347238T_real(F,A) = zero_zero_real )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A))
             => ( aa_VEBT_VEBT_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7147_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A) = zero_zero_real )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A))
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7148_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A) = zero_zero_real )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A))
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7149_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X4))) )
       => ( ( groups1270011288395367621r_real(F,A) = zero_zero_real )
        <=> ! [X3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
             => ( aa_Code_integer_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7150_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A) = zero_zero_real )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A))
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7151_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
       => ( ( groups136491112297645522BT_rat(F,A) = zero_zero_rat )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A))
             => ( aa_VEBT_VEBT_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7152_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( ( groups1300246762558778688al_rat(F,A) = zero_zero_rat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A))
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7153_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( ( groups2906978787729119204at_rat(F,A) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A))
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7154_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( ( groups3906332499630173760nt_rat(F,A) = zero_zero_rat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A))
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7155_sum__nonneg__eq__0__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X4))) )
       => ( ( groups6602215022474089585er_rat(F,A) = zero_zero_rat )
        <=> ! [X3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
             => ( aa_Code_integer_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7156_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X6: int] :
              ( pp(aa_set_int_bool(member_int(X6),A))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X6)),aa_int_real(G,X6))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A)),groups8778361861064173332t_real(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7157_sum__strict__mono__ex1,axiom,
    ! [A: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Code_integer_real(F,X4)),aa_Code_integer_real(G,X4))) )
       => ( ? [X6: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Code_integer_real(F,X6)),aa_Code_integer_real(G,X6))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1270011288395367621r_real(F,A)),groups1270011288395367621r_real(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7158_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X6: complex] :
              ( pp(aa_set_complex_bool(member_complex(X6),A))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X6)),aa_complex_real(G,X6))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7159_sum__strict__mono__ex1,axiom,
    ! [A: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
       => ( ? [X6: nat] :
              ( pp(aa_set_nat_bool(member_nat(X6),A))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X6)),aa_nat_rat(G,X6))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7160_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
       => ( ? [X6: int] :
              ( pp(aa_set_int_bool(member_int(X6),A))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X6)),aa_int_rat(G,X6))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A)),groups3906332499630173760nt_rat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7161_sum__strict__mono__ex1,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,X4)),aa_Code_integer_rat(G,X4))) )
       => ( ? [X6: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X6)),aa_Code_integer_rat(G,X6))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups6602215022474089585er_rat(F,A)),groups6602215022474089585er_rat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7162_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
       => ( ? [X6: complex] :
              ( pp(aa_set_complex_bool(member_complex(X6),A))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X6)),aa_complex_rat(G,X6))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7163_sum__strict__mono__ex1,axiom,
    ! [A: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X6: int] :
              ( pp(aa_set_int_bool(member_int(X6),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X6)),aa_int_nat(G,X6))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7164_sum__strict__mono__ex1,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Code_integer_nat(F,X4)),aa_Code_integer_nat(G,X4))) )
       => ( ? [X6: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X6),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,X6)),aa_Code_integer_nat(G,X6))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups7237345082560585321er_nat(F,A)),groups7237345082560585321er_nat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7165_sum__strict__mono__ex1,axiom,
    ! [A: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X6: complex] :
              ( pp(aa_set_complex_bool(member_complex(X6),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X6)),aa_complex_nat(G,X6))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(G,A))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7166_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y22)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y1)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups2073611262835488442omplex(H,S)),groups2073611262835488442omplex(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7167_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y22)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y1)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_int_complex(H,X4)),aa_int_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups3049146728041665814omplex(H,S)),groups3049146728041665814omplex(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7168_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S: set_Code_integer,H: fun_Co1900716531650368264omplex,G: fun_Co1900716531650368264omplex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y22)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y1)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X23),Y22))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_Cod5621350732091595593omplex(H,X4)),aa_Cod5621350732091595593omplex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups8024822376189712711omplex(H,S)),groups8024822376189712711omplex(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7169_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups8778361861064173332t_real(H,S)),groups8778361861064173332t_real(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7170_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y22))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Code_integer_real(H,X4)),aa_Code_integer_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups1270011288395367621r_real(H,S)),groups1270011288395367621r_real(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7171_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups5808333547571424918x_real(H,S)),groups5808333547571424918x_real(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7172_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y1)),aa_rat_rat(plus_plus_rat(X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups2906978787729119204at_rat(H,S)),groups2906978787729119204at_rat(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7173_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y1)),aa_rat_rat(plus_plus_rat(X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups3906332499630173760nt_rat(H,S)),groups3906332499630173760nt_rat(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7174_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_Code_integer,H: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y1)),aa_rat_rat(plus_plus_rat(X23),Y22))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_Code_integer_rat(H,X4)),aa_Code_integer_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups6602215022474089585er_rat(H,S)),groups6602215022474089585er_rat(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7175_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y1)),aa_rat_rat(plus_plus_rat(X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups5058264527183730370ex_rat(H,S)),groups5058264527183730370ex_rat(G,S))) ) ) ) ) ).

% sum.related
tff(fact_7176_sum__strict__mono,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( A != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_VEBT_VEBT_real(F,X4)),aa_VEBT_VEBT_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A)),groups2240296850493347238T_real(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7177_sum__strict__mono,axiom,
    ! [A: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( A != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A)),groups8097168146408367636l_real(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7178_sum__strict__mono,axiom,
    ! [A: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Code_integer_real(F,X4)),aa_Code_integer_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1270011288395367621r_real(F,A)),groups1270011288395367621r_real(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7179_sum__strict__mono,axiom,
    ! [A: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( A != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7180_sum__strict__mono,axiom,
    ! [A: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A)),groups8778361861064173332t_real(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7181_sum__strict__mono,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( A != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_VEBT_VEBT_rat(F,X4)),aa_VEBT_VEBT_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A)),groups136491112297645522BT_rat(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7182_sum__strict__mono,axiom,
    ! [A: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( A != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A)),groups1300246762558778688al_rat(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7183_sum__strict__mono,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X4)),aa_Code_integer_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups6602215022474089585er_rat(F,A)),groups6602215022474089585er_rat(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7184_sum__strict__mono,axiom,
    ! [A: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( A != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7185_sum__strict__mono,axiom,
    ! [A: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(G,A))) ) ) ) ).

% sum_strict_mono
tff(fact_7186_sum_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A)) = groups2240296850493347238T_real(G,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7187_sum_Oinsert__if,axiom,
    ! [A: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( pp(aa_set_real_bool(member_real(X2),A))
         => ( groups8097168146408367636l_real(G,insert_real(X2,A)) = groups8097168146408367636l_real(G,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A))
         => ( groups8097168146408367636l_real(G,insert_real(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7188_sum_Oinsert__if,axiom,
    ! [A: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( pp(aa_set_int_bool(member_int(X2),A))
         => ( groups8778361861064173332t_real(G,insert_int(X2,A)) = groups8778361861064173332t_real(G,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A))
         => ( groups8778361861064173332t_real(G,insert_int(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X2)),groups8778361861064173332t_real(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7189_sum_Oinsert__if,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
         => ( groups1270011288395367621r_real(G,insert_Code_integer(X2,A)) = groups1270011288395367621r_real(G,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
         => ( groups1270011288395367621r_real(G,insert_Code_integer(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,X2)),groups1270011288395367621r_real(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7190_sum_Oinsert__if,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( pp(aa_set_complex_bool(member_complex(X2),A))
         => ( groups5808333547571424918x_real(G,insert_complex(X2,A)) = groups5808333547571424918x_real(G,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X2),A))
         => ( groups5808333547571424918x_real(G,insert_complex(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7191_sum_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A)) = groups136491112297645522BT_rat(G,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7192_sum_Oinsert__if,axiom,
    ! [A: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( pp(aa_set_real_bool(member_real(X2),A))
         => ( groups1300246762558778688al_rat(G,insert_real(X2,A)) = groups1300246762558778688al_rat(G,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A))
         => ( groups1300246762558778688al_rat(G,insert_real(X2,A)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7193_sum_Oinsert__if,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( pp(aa_set_nat_bool(member_nat(X2),A))
         => ( groups2906978787729119204at_rat(G,insert_nat(X2,A)) = groups2906978787729119204at_rat(G,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
         => ( groups2906978787729119204at_rat(G,insert_nat(X2,A)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,X2)),groups2906978787729119204at_rat(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7194_sum_Oinsert__if,axiom,
    ! [A: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( pp(aa_set_int_bool(member_int(X2),A))
         => ( groups3906332499630173760nt_rat(G,insert_int(X2,A)) = groups3906332499630173760nt_rat(G,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A))
         => ( groups3906332499630173760nt_rat(G,insert_int(X2,A)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X2)),groups3906332499630173760nt_rat(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7195_sum_Oinsert__if,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
         => ( groups6602215022474089585er_rat(G,insert_Code_integer(X2,A)) = groups6602215022474089585er_rat(G,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
         => ( groups6602215022474089585er_rat(G,insert_Code_integer(X2,A)) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,X2)),groups6602215022474089585er_rat(G,A)) ) ) ) ) ).

% sum.insert_if
tff(fact_7196_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_VEBT_VEBT,S: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T5: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T5,T6))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),T6))
                       => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_VEBT_VEBT_complex(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7197_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_real,S: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T5: set_real,G: fun_VE5569070535936029429omplex,H: fun_real_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set_real_bool(finite_finite_real,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T5,T6))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T6))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_real_complex(H,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7198_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_VEBT_VEBT,S: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T5: set_VEBT_VEBT,G: fun_real_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T5,T6))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),T6))
                       => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_VEBT_VEBT_complex(H,aa_real_VEBT_VEBT(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7199_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_real,S: set_real,I: fun_real_real,J: fun_real_real,T5: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set_real_bool(finite_finite_real,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T5,T6))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T6))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7200_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_int,S: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T5: set_int,G: fun_VE5569070535936029429omplex,H: fun_int_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set_int_bool(finite_finite_int,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T5,T6))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T6))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_int_complex(H,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S) = groups3049146728041665814omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7201_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_int,S: set_real,I: fun_int_real,J: fun_real_int,T5: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set_int_bool(finite_finite_int,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T5,T6))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T6))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S) = groups3049146728041665814omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7202_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_Code_integer,S: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T5: set_Code_integer,G: fun_VE5569070535936029429omplex,H: fun_Co1900716531650368264omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set1374477347934578310r_bool(member_Code_integer(aa_VEB5935388676523804373nteger(J,A3)),minus_2355218937544613996nteger(T5,T6))) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),T6))
                       => ( aa_Cod5621350732091595593omplex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_Cod5621350732091595593omplex(H,aa_VEB5935388676523804373nteger(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S) = groups8024822376189712711omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7203_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_Code_integer,S: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T5: set_Code_integer,G: fun_real_complex,H: fun_Co1900716531650368264omplex] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set1374477347934578310r_bool(member_Code_integer(aa_real_Code_integer(J,A3)),minus_2355218937544613996nteger(T5,T6))) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B3)) = B3 ) )
             => ( ! [B3: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_Code_integer_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),T6))
                       => ( aa_Cod5621350732091595593omplex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_Cod5621350732091595593omplex(H,aa_real_Code_integer(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S) = groups8024822376189712711omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7204_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T6: set_VEBT_VEBT,S: set_int,I: fun_VEBT_VEBT_int,J: fun_int_VEBT_VEBT,T5: set_VEBT_VEBT,G: fun_int_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S6))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T6))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S,S6)))
             => ( aa_VEBT_VEBT_int(I,aa_int_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S,S6)))
               => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_int_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T5,T6))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                 => ( aa_int_VEBT_VEBT(J,aa_VEBT_VEBT_int(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                   => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(I,B3)),minus_minus_set_int(S,S6))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S6))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),T6))
                       => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S))
                         => ( aa_VEBT_VEBT_complex(H,aa_int_VEBT_VEBT(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7205_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T6: set_real,S: set_int,I: fun_real_int,J: fun_int_real,T5: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S6))
     => ( pp(aa_set_real_bool(finite_finite_real,T6))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S,S6)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S,S6)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T5,T6))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S,S6))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S6))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T6))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S))
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7206_sum__nonneg__0,axiom,
    ! [S2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2))) )
       => ( ( groups2240296850493347238T_real(F,S2) = zero_zero_real )
         => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),S2))
           => ( aa_VEBT_VEBT_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7207_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7208_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7209_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,F: fun_Co3960896333048504454r_real,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2))) )
       => ( ( groups1270011288395367621r_real(F,S2) = zero_zero_real )
         => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),S2))
           => ( aa_Code_integer_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7210_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S2) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I),S2))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7211_sum__nonneg__0,axiom,
    ! [S2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2))) )
       => ( ( groups136491112297645522BT_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),S2))
           => ( aa_VEBT_VEBT_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7212_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7213_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I),S2))
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7214_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7215_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I2))) )
       => ( ( groups6602215022474089585er_rat(F,S2) = zero_zero_rat )
         => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),S2))
           => ( aa_Code_integer_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7216_sum__nonneg__leq__bound,axiom,
    ! [S2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B: real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2))) )
       => ( ( groups2240296850493347238T_real(F,S2) = B )
         => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7217_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_real,B: real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S2) = B )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7218_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_real,B: real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S2) = B )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7219_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: fun_Co3960896333048504454r_real,B: real,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2))) )
       => ( ( groups1270011288395367621r_real(F,S2) = B )
         => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Code_integer_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7220_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_real,B: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S2) = B )
         => ( pp(aa_set_complex_bool(member_complex(I),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7221_sum__nonneg__leq__bound,axiom,
    ! [S2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,B: rat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2))) )
       => ( ( groups136491112297645522BT_rat(F,S2) = B )
         => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7222_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_rat,B: rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S2) = B )
         => ( pp(aa_set_real_bool(member_real(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7223_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_rat,B: rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S2) = B )
         => ( pp(aa_set_nat_bool(member_nat(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7224_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_rat,B: rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = B )
         => ( pp(aa_set_int_bool(member_int(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7225_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat,B: rat,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I2))) )
       => ( ( groups6602215022474089585er_rat(F,S2) = B )
         => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,I)),B)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7226_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A,collect_int(aTP_Lamm_pf(G)))) = groups3049146728041665814omplex(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7227_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups8024822376189712711omplex(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_pg(G)))) = groups8024822376189712711omplex(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7228_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A,collect_int(aTP_Lamm_ph(G)))) = groups8778361861064173332t_real(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7229_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_pi(G)))) = groups1270011288395367621r_real(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7230_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_pj(G)))) = groups5808333547571424918x_real(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7231_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A,collect_int(aTP_Lamm_pk(G)))) = groups3906332499630173760nt_rat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7232_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_pl(G)))) = groups6602215022474089585er_rat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7233_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_pm(G)))) = groups5058264527183730370ex_rat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7234_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A,collect_int(aTP_Lamm_pn(G)))) = groups4541462559716669496nt_nat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7235_sum_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_po(G)))) = groups7237345082560585321er_nat(G,A) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7236_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),set_or4665077453230672383an_nat(M,N)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_7237_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_pq(G),set_or4665077453230672383an_nat(M,N)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_7238_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7239_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_pq(G),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7240_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr(G),K)),set_or4665077453230672383an_nat(M,N)) ).

% sum.shift_bounds_nat_ivl
tff(fact_7241_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ps(G),K),set_or4665077453230672383an_nat(M,N)) ).

% sum.shift_bounds_nat_ivl
tff(fact_7242_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr(G),K)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7243_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ps(G),K),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7244_sum__pos2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I)))
         => ( ! [I2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7245_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7246_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7247_sum__pos2,axiom,
    ! [I6: set_Code_integer,I: code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Code_integer_real(F,I)))
         => ( ! [I2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups1270011288395367621r_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7248_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7249_sum__pos2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7250_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7251_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7252_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7253_sum__pos2,axiom,
    ! [I6: set_Code_integer,I: code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,I)))
         => ( ! [I2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_7254_sum__pos,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7255_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7256_sum__pos,axiom,
    ! [I6: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( ( I6 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Code_integer_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups1270011288395367621r_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7257_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7258_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7259_sum__pos,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7260_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7261_sum__pos,axiom,
    ! [I6: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( ( I6 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7262_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7263_sum__pos,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_7264_sum_Osame__carrier,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,A) = groups1794756597179926696omplex(H,B) )
              <=> ( groups1794756597179926696omplex(G,C) = groups1794756597179926696omplex(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7265_sum_Osame__carrier,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A) = groups5754745047067104278omplex(H,B) )
              <=> ( groups5754745047067104278omplex(G,C) = groups5754745047067104278omplex(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7266_sum_Osame__carrier,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co1900716531650368264omplex,H: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Cod5621350732091595593omplex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Cod5621350732091595593omplex(H,B3) = zero_zero_complex ) )
             => ( ( groups8024822376189712711omplex(G,A) = groups8024822376189712711omplex(H,B) )
              <=> ( groups8024822376189712711omplex(G,C) = groups8024822376189712711omplex(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7267_sum_Osame__carrier,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_real(H,B3) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A) = groups2240296850493347238T_real(H,B) )
              <=> ( groups2240296850493347238T_real(G,C) = groups2240296850493347238T_real(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7268_sum_Osame__carrier,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A) = groups8097168146408367636l_real(H,B) )
              <=> ( groups8097168146408367636l_real(G,C) = groups8097168146408367636l_real(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7269_sum_Osame__carrier,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_real(H,B3) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,A) = groups1270011288395367621r_real(H,B) )
              <=> ( groups1270011288395367621r_real(G,C) = groups1270011288395367621r_real(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7270_sum_Osame__carrier,axiom,
    ! [C: set_complex,A: set_complex,B: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),C))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C,A)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C,B)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A) = groups5808333547571424918x_real(H,B) )
              <=> ( groups5808333547571424918x_real(G,C) = groups5808333547571424918x_real(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7271_sum_Osame__carrier,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A) = groups136491112297645522BT_rat(H,B) )
              <=> ( groups136491112297645522BT_rat(G,C) = groups136491112297645522BT_rat(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7272_sum_Osame__carrier,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A) = groups1300246762558778688al_rat(H,B) )
              <=> ( groups1300246762558778688al_rat(G,C) = groups1300246762558778688al_rat(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7273_sum_Osame__carrier,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(G,A) = groups6602215022474089585er_rat(H,B) )
              <=> ( groups6602215022474089585er_rat(G,C) = groups6602215022474089585er_rat(H,C) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7274_sum_Osame__carrierI,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,C) = groups1794756597179926696omplex(H,C) )
               => ( groups1794756597179926696omplex(G,A) = groups1794756597179926696omplex(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7275_sum_Osame__carrierI,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C) = groups5754745047067104278omplex(H,C) )
               => ( groups5754745047067104278omplex(G,A) = groups5754745047067104278omplex(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7276_sum_Osame__carrierI,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co1900716531650368264omplex,H: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Cod5621350732091595593omplex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Cod5621350732091595593omplex(H,B3) = zero_zero_complex ) )
             => ( ( groups8024822376189712711omplex(G,C) = groups8024822376189712711omplex(H,C) )
               => ( groups8024822376189712711omplex(G,A) = groups8024822376189712711omplex(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7277_sum_Osame__carrierI,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_real(H,B3) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C) = groups2240296850493347238T_real(H,C) )
               => ( groups2240296850493347238T_real(G,A) = groups2240296850493347238T_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7278_sum_Osame__carrierI,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C) = groups8097168146408367636l_real(H,C) )
               => ( groups8097168146408367636l_real(G,A) = groups8097168146408367636l_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7279_sum_Osame__carrierI,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_real(H,B3) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,C) = groups1270011288395367621r_real(H,C) )
               => ( groups1270011288395367621r_real(G,A) = groups1270011288395367621r_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7280_sum_Osame__carrierI,axiom,
    ! [C: set_complex,A: set_complex,B: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),C))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C,A)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C,B)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C) = groups5808333547571424918x_real(H,C) )
               => ( groups5808333547571424918x_real(G,A) = groups5808333547571424918x_real(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7281_sum_Osame__carrierI,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C) = groups136491112297645522BT_rat(H,C) )
               => ( groups136491112297645522BT_rat(G,A) = groups136491112297645522BT_rat(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7282_sum_Osame__carrierI,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C) = groups1300246762558778688al_rat(H,C) )
               => ( groups1300246762558778688al_rat(G,A) = groups1300246762558778688al_rat(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7283_sum_Osame__carrierI,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(G,C) = groups6602215022474089585er_rat(H,C) )
               => ( groups6602215022474089585er_rat(G,A) = groups6602215022474089585er_rat(H,B) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7284_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Cod5621350732091595593omplex(G,X4) = zero_zero_complex ) )
         => ( groups8024822376189712711omplex(G,S) = groups8024822376189712711omplex(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7285_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(G,X4) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,S) = groups1270011288395367621r_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7286_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S) = groups5808333547571424918x_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7287_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_rat(G,X4) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,S) = groups6602215022474089585er_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7288_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S) = groups5058264527183730370ex_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7289_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_nat(G,X4) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,S) = groups7237345082560585321er_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7290_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S) = groups5693394587270226106ex_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7291_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_int(G,X4) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,S) = groups7234854612051535045er_int(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7292_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S) = groups5690904116761175830ex_int(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7293_sum_Omono__neutral__left,axiom,
    ! [T5: set_nat,S: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T5))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S),T5))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T5,S)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S) = groups2073611262835488442omplex(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7294_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Cod5621350732091595593omplex(G,X4) = zero_zero_complex ) )
         => ( groups8024822376189712711omplex(G,T5) = groups8024822376189712711omplex(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7295_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(G,X4) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,T5) = groups1270011288395367621r_real(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7296_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T5) = groups5808333547571424918x_real(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7297_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_rat(G,X4) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,T5) = groups6602215022474089585er_rat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7298_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T5) = groups5058264527183730370ex_rat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7299_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_nat(G,X4) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,T5) = groups7237345082560585321er_nat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7300_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T5) = groups5693394587270226106ex_nat(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7301_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_int(G,X4) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,T5) = groups7234854612051535045er_int(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7302_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T5) = groups5690904116761175830ex_int(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7303_sum_Omono__neutral__right,axiom,
    ! [T5: set_nat,S: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T5))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S),T5))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T5,S)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T5) = groups2073611262835488442omplex(G,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7304_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,H: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_complex(G,X4) = aa_VEBT_VEBT_complex(H,X4) ) )
           => ( groups1794756597179926696omplex(G,S) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7305_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,S) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7306_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,H: fun_Co1900716531650368264omplex,G: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Cod5621350732091595593omplex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Cod5621350732091595593omplex(G,X4) = aa_Cod5621350732091595593omplex(H,X4) ) )
           => ( groups8024822376189712711omplex(G,S) = groups8024822376189712711omplex(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7307_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2240296850493347238T_real(G,S) = groups2240296850493347238T_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7308_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S) = groups8097168146408367636l_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7309_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_real(G,X4) = aa_Code_integer_real(H,X4) ) )
           => ( groups1270011288395367621r_real(G,S) = groups1270011288395367621r_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7310_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S) = groups5808333547571424918x_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7311_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups136491112297645522BT_rat(G,S) = groups136491112297645522BT_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7312_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,S) = groups1300246762558778688al_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7313_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,H: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_rat(G,X4) = aa_Code_integer_rat(H,X4) ) )
           => ( groups6602215022474089585er_rat(G,S) = groups6602215022474089585er_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7314_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_complex(G,X4) = aa_VEBT_VEBT_complex(H,X4) ) )
           => ( groups1794756597179926696omplex(G,T5) = groups1794756597179926696omplex(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7315_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,T5) = groups5754745047067104278omplex(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7316_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co1900716531650368264omplex,H: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Cod5621350732091595593omplex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Cod5621350732091595593omplex(G,X4) = aa_Cod5621350732091595593omplex(H,X4) ) )
           => ( groups8024822376189712711omplex(G,T5) = groups8024822376189712711omplex(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7317_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2240296850493347238T_real(G,T5) = groups2240296850493347238T_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7318_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T5) = groups8097168146408367636l_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7319_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_real(G,X4) = aa_Code_integer_real(H,X4) ) )
           => ( groups1270011288395367621r_real(G,T5) = groups1270011288395367621r_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7320_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T5) = groups5808333547571424918x_real(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7321_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups136491112297645522BT_rat(G,T5) = groups136491112297645522BT_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7322_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,T5) = groups1300246762558778688al_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7323_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_rat(G,X4) = aa_Code_integer_rat(H,X4) ) )
           => ( groups6602215022474089585er_rat(G,T5) = groups6602215022474089585er_rat(H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7324_sum_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups1270011288395367621r_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A,B))),groups1270011288395367621r_real(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7325_sum_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups5808333547571424918x_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,B))),groups5808333547571424918x_real(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7326_sum_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups6602215022474089585er_rat(G,A) = aa_rat_rat(plus_plus_rat(groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A,B))),groups6602215022474089585er_rat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7327_sum_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups5058264527183730370ex_rat(G,A) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,B))),groups5058264527183730370ex_rat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7328_sum_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups7237345082560585321er_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A,B))),groups7237345082560585321er_nat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7329_sum_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups5693394587270226106ex_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A,B))),groups5693394587270226106ex_nat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7330_sum_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups7234854612051535045er_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A,B))),groups7234854612051535045er_int(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7331_sum_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups5690904116761175830ex_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A,B))),groups5690904116761175830ex_int(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7332_sum_Osubset__diff,axiom,
    ! [B: set_nat,A: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A))
       => ( groups2906978787729119204at_rat(G,A) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A,B))),groups2906978787729119204at_rat(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7333_sum_Osubset__diff,axiom,
    ! [B: set_nat,A: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A))
       => ( groups3539618377306564664at_int(G,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A,B))),groups3539618377306564664at_int(G,B)) ) ) ) ).

% sum.subset_diff
tff(fact_7334_sum__diff,axiom,
    ! [A: set_Code_integer,B: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
       => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A,B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1270011288395367621r_real(F,A)),groups1270011288395367621r_real(F,B)) ) ) ) ).

% sum_diff
tff(fact_7335_sum__diff,axiom,
    ! [A: set_complex,B: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A,B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(F,B)) ) ) ) ).

% sum_diff
tff(fact_7336_sum__diff,axiom,
    ! [A: set_Code_integer,B: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
       => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A,B)) = aa_rat_rat(minus_minus_rat(groups6602215022474089585er_rat(F,A)),groups6602215022474089585er_rat(F,B)) ) ) ) ).

% sum_diff
tff(fact_7337_sum__diff,axiom,
    ! [A: set_complex,B: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A,B)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(F,B)) ) ) ) ).

% sum_diff
tff(fact_7338_sum__diff,axiom,
    ! [A: set_Code_integer,B: set_Code_integer,F: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
       => ( groups7234854612051535045er_int(F,minus_2355218937544613996nteger(A,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups7234854612051535045er_int(F,A)),groups7234854612051535045er_int(F,B)) ) ) ) ).

% sum_diff
tff(fact_7339_sum__diff,axiom,
    ! [A: set_complex,B: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A)),groups5690904116761175830ex_int(F,B)) ) ) ) ).

% sum_diff
tff(fact_7340_sum__diff,axiom,
    ! [A: set_nat,B: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A,B)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(F,B)) ) ) ) ).

% sum_diff
tff(fact_7341_sum__diff,axiom,
    ! [A: set_nat,B: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A)),groups3539618377306564664at_int(F,B)) ) ) ) ).

% sum_diff
tff(fact_7342_sum__diff,axiom,
    ! [A: set_int,B: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A,B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A)),groups8778361861064173332t_real(F,B)) ) ) ) ).

% sum_diff
tff(fact_7343_sum__diff,axiom,
    ! [A: set_int,B: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A,B)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A)),groups3906332499630173760nt_rat(F,B)) ) ) ) ).

% sum_diff
tff(fact_7344_sum_Oivl__cong,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A2 = C2 )
     => ( ( B2 = D2 )
       => ( ! [X4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X4))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D2))
               => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(A2,B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(H),set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_7345_sum_Oivl__cong,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( ( A2 = C2 )
     => ( ( B2 = D2 )
       => ( ! [X4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X4))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D2))
               => ( aa_nat_real(G,X4) = aa_nat_real(H,X4) ) ) )
         => ( groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(A2,B2)) = groups6591440286371151544t_real(H,set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_7346_sum_Oivl__cong,axiom,
    ! [A2: int,C2: int,B2: int,D2: int,G: fun_int_int,H: fun_int_int] :
      ( ( A2 = C2 )
     => ( ( B2 = D2 )
       => ( ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),X4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D2))
               => ( aa_int_int(G,X4) = aa_int_int(H,X4) ) ) )
         => ( groups4538972089207619220nt_int(G,set_or4662586982721622107an_int(A2,B2)) = groups4538972089207619220nt_int(H,set_or4662586982721622107an_int(C2,D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_7347_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,N))),groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(N,P2))) = groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_7348_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,N))),groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(N,P2))) = groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_7349_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(N,P2))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_7350_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,N))),groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(N,P2))) = groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_7351_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P2: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,set_or4665077453230672383an_nat(M,P2))),groups2906978787729119204at_rat(F,set_or4665077453230672383an_nat(M,N))) = groups2906978787729119204at_rat(F,set_or4665077453230672383an_nat(N,P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_7352_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P2: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,set_or4665077453230672383an_nat(M,P2))),groups3539618377306564664at_int(F,set_or4665077453230672383an_nat(M,N))) = groups3539618377306564664at_int(F,set_or4665077453230672383an_nat(N,P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_7353_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P2: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,set_or4665077453230672383an_nat(M,P2))),groups6591440286371151544t_real(F,set_or4665077453230672383an_nat(M,N))) = groups6591440286371151544t_real(F,set_or4665077453230672383an_nat(N,P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_7354_sum__power__add,axiom,
    ! [X2: complex,M: nat,I6: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pt(X2),M),I6) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),M)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),I6)) ).

% sum_power_add
tff(fact_7355_sum__power__add,axiom,
    ! [X2: code_integer,M: nat,I6: set_nat] : groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_pu(X2),M),I6) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),M)),groups7501900531339628137nteger(power_8256067586552552935nteger(X2),I6)) ).

% sum_power_add
tff(fact_7356_sum__power__add,axiom,
    ! [X2: rat,M: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_pv(X2),M),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),M)),groups2906978787729119204at_rat(power_power_rat(X2),I6)) ).

% sum_power_add
tff(fact_7357_sum__power__add,axiom,
    ! [X2: int,M: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pw(X2),M),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),M)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),I6)) ).

% sum_power_add
tff(fact_7358_sum__power__add,axiom,
    ! [X2: real,M: nat,I6: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_px(X2),M),I6) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),I6)) ).

% sum_power_add
tff(fact_7359_sum__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A)),groups2240296850493347238T_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7360_sum__mono2,axiom,
    ! [B: set_real,A: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A)),groups8097168146408367636l_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7361_sum__mono2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( ! [B3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1270011288395367621r_real(F,A)),groups1270011288395367621r_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7362_sum__mono2,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7363_sum__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A)),groups136491112297645522BT_rat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7364_sum__mono2,axiom,
    ! [B: set_real,A: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A)),groups1300246762558778688al_rat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7365_sum__mono2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( ! [B3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups6602215022474089585er_rat(F,A)),groups6602215022474089585er_rat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7366_sum__mono2,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7367_sum__mono2,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7368_sum__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,A)),groups771621172384141258BT_nat(F,B))) ) ) ) ).

% sum_mono2
tff(fact_7369_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(N,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(G),N),M)),set_or1269000886237332187st_nat(N,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7370_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(N,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_pz(G),N),M),set_or1269000886237332187st_nat(N,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7371_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7372_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Co3960896333048504454r_real,X2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups1270011288395367621r_real(G,insert_Code_integer(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,X2)),groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_7373_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_real,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups5808333547571424918x_real(G,insert_complex(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7374_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7375_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,X2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups6602215022474089585er_rat(G,insert_Code_integer(X2,A)) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,X2)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_7376_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_rat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups5058264527183730370ex_rat(G,insert_complex(X2,A)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X2)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7377_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X2,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X2)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7378_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_nat,X2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups7237345082560585321er_nat(G,insert_Code_integer(X2,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(G,X2)),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_7379_sum_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_nat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups5693394587270226106ex_nat(G,insert_complex(X2,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X2)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7380_sum_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups769130701875090982BT_int(G,insert_VEBT_VEBT(X2,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,X2)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_7381_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups2240296850493347238T_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7382_sum_Oremove,axiom,
    ! [A: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => ( groups8097168146408367636l_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,minus_minus_set_real(A,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7383_sum_Oremove,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups1270011288395367621r_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,X2)),groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_7384_sum_Oremove,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(member_complex(X2),A))
       => ( groups5808333547571424918x_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7385_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups136491112297645522BT_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7386_sum_Oremove,axiom,
    ! [A: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => ( groups1300246762558778688al_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,minus_minus_set_real(A,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7387_sum_Oremove,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups6602215022474089585er_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,X2)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_7388_sum_Oremove,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(member_complex(X2),A))
       => ( groups5058264527183730370ex_rat(G,A) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X2)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7389_sum_Oremove,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups771621172384141258BT_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X2)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_7390_sum_Oremove,axiom,
    ! [A: set_real,X2: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => ( groups1935376822645274424al_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X2)),groups1935376822645274424al_nat(G,minus_minus_set_real(A,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7391_sum__diff1,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
         => ( groups2240296850493347238T_real(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2240296850493347238T_real(F,A)),aa_VEBT_VEBT_real(F,A2)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
         => ( groups2240296850493347238T_real(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = groups2240296850493347238T_real(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7392_sum__diff1,axiom,
    ! [A: set_real,A2: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( pp(aa_set_real_bool(member_real(A2),A))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A)),aa_real_real(F,A2)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),A))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = groups8097168146408367636l_real(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7393_sum__diff1,axiom,
    ! [A: set_Code_integer,A2: code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
         => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1270011288395367621r_real(F,A)),aa_Code_integer_real(F,A2)) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
         => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = groups1270011288395367621r_real(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7394_sum__diff1,axiom,
    ! [A: set_complex,A2: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),A))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A)),aa_complex_real(F,A2)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),A))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = groups5808333547571424918x_real(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7395_sum__diff1,axiom,
    ! [A: set_int,A2: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( pp(aa_set_int_bool(member_int(A2),A))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A)),aa_int_real(F,A2)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),A))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = groups8778361861064173332t_real(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7396_sum__diff1,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
         => ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = aa_rat_rat(minus_minus_rat(groups136491112297645522BT_rat(F,A)),aa_VEBT_VEBT_rat(F,A2)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
         => ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = groups136491112297645522BT_rat(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7397_sum__diff1,axiom,
    ! [A: set_real,A2: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( pp(aa_set_real_bool(member_real(A2),A))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(F,A)),aa_real_rat(F,A2)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),A))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = groups1300246762558778688al_rat(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7398_sum__diff1,axiom,
    ! [A: set_Code_integer,A2: code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
         => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = aa_rat_rat(minus_minus_rat(groups6602215022474089585er_rat(F,A)),aa_Code_integer_rat(F,A2)) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
         => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = groups6602215022474089585er_rat(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7399_sum__diff1,axiom,
    ! [A: set_complex,A2: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),A))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A)),aa_complex_rat(F,A2)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),A))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = groups5058264527183730370ex_rat(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7400_sum__diff1,axiom,
    ! [A: set_int,A2: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( pp(aa_set_int_bool(member_int(A2),A))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A)),aa_int_rat(F,A2)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),A))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = groups3906332499630173760nt_rat(F,A) ) ) ) ) ).

% sum_diff1
tff(fact_7401_sum_Odelta__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real,C2: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_qa(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B2,A2)),groups2240296850493347238T_real(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_qa(A2),B2),C2),S) = groups2240296850493347238T_real(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7402_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B2: fun_real_real,C2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qb(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B2,A2)),groups8097168146408367636l_real(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qb(A2),B2),C2),S) = groups8097168146408367636l_real(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7403_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real,C2: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_qc(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(B2,A2)),groups1270011288395367621r_real(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_qc(A2),B2),C2),S) = groups1270011288395367621r_real(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7404_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_real,C2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_qd(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B2,A2)),groups5808333547571424918x_real(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_qd(A2),B2),C2),S) = groups5808333547571424918x_real(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7405_sum_Odelta__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_rat,C2: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_qe(A2),B2),C2),S) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(B2,A2)),groups136491112297645522BT_rat(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_qe(A2),B2),C2),S) = groups136491112297645522BT_rat(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7406_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B2: fun_real_rat,C2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_qf(A2),B2),C2),S) = aa_rat_rat(plus_plus_rat(aa_real_rat(B2,A2)),groups1300246762558778688al_rat(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_qf(A2),B2),C2),S) = groups1300246762558778688al_rat(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7407_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Code_integer_rat,C2: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_qg(A2),B2),C2),S) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(B2,A2)),groups6602215022474089585er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_qg(A2),B2),C2),S) = groups6602215022474089585er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7408_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_rat,C2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_qh(A2),B2),C2),S) = aa_rat_rat(plus_plus_rat(aa_complex_rat(B2,A2)),groups5058264527183730370ex_rat(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_qh(A2),B2),C2),S) = groups5058264527183730370ex_rat(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7409_sum_Odelta__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_nat,C2: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_qi(A2),B2),C2),S) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B2,A2)),groups771621172384141258BT_nat(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_qi(A2),B2),C2),S) = groups771621172384141258BT_nat(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7410_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B2: fun_real_nat,C2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_qj(A2),B2),C2),S) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B2,A2)),groups1935376822645274424al_nat(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_qj(A2),B2),C2),S) = groups1935376822645274424al_nat(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_7411_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2073611262835488442omplex(F,set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_7412_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_7413_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_7414_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_7415_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups6591440286371151544t_real(F,set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_7416_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7417_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7418_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7419_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7420_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7421_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_rat(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_7422_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_int(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_7423_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_nat(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_7424_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_real(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_7425_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7426_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7427_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7428_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_7429_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7430_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7431_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7432_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7433_sum_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_rat(G,B2)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7434_sum_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_int(G,B2)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7435_sum_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(A2,B2))),aa_nat_nat(G,B2)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7436_sum_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_real(G,B2)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_7437_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7438_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7439_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7440_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7441_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7442_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7443_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7444_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7445_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,N)),groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% sum.last_plus
tff(fact_7446_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,N)),groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% sum.last_plus
tff(fact_7447_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,N)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,N))) ) ) ).

% sum.last_plus
tff(fact_7448_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,N)),groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% sum.last_plus
tff(fact_7449_sum__strict__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B,A)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,B2)))
           => ( ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A)),groups2240296850493347238T_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7450_sum__strict__mono2,axiom,
    ! [B: set_real,A: set_real,B2: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( pp(aa_set_real_bool(member_real(B2),minus_minus_set_real(B,A)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B2)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A)),groups8097168146408367636l_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7451_sum__strict__mono2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,B2: code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B2),minus_2355218937544613996nteger(B,A)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Code_integer_real(F,B2)))
           => ( ! [X4: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1270011288395367621r_real(F,A)),groups1270011288395367621r_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7452_sum__strict__mono2,axiom,
    ! [B: set_complex,A: set_complex,B2: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( pp(aa_set_complex_bool(member_complex(B2),minus_811609699411566653omplex(B,A)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B2)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A)),groups5808333547571424918x_real(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7453_sum__strict__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B,A)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B2)))
           => ( ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A)),groups136491112297645522BT_rat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7454_sum__strict__mono2,axiom,
    ! [B: set_real,A: set_real,B2: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( pp(aa_set_real_bool(member_real(B2),minus_minus_set_real(B,A)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B2)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A)),groups1300246762558778688al_rat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7455_sum__strict__mono2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,B2: code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B2),minus_2355218937544613996nteger(B,A)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,B2)))
           => ( ! [X4: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),B))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups6602215022474089585er_rat(F,A)),groups6602215022474089585er_rat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7456_sum__strict__mono2,axiom,
    ! [B: set_complex,A: set_complex,B2: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( pp(aa_set_complex_bool(member_complex(B2),minus_811609699411566653omplex(B,A)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B2)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A)),groups5058264527183730370ex_rat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7457_sum__strict__mono2,axiom,
    ! [B: set_nat,A: set_nat,B2: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( pp(aa_set_nat_bool(member_nat(B2),minus_minus_set_nat(B,A)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B2)))
           => ( ! [X4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X4),B))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A)),groups2906978787729119204at_rat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7458_sum__strict__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B,A)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B2)))
           => ( ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups771621172384141258BT_nat(F,A)),groups771621172384141258BT_nat(F,B))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7459_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
       => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),groups2240296850493347238T_real(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7460_member__le__sum,axiom,
    ! [I: real,A: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I),A))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A,insert_real(I,bot_bot_set_real))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7461_member__le__sum,axiom,
    ! [I: code_integer,A: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(A,insert_Code_integer(I,bot_bo3990330152332043303nteger))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X4))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Code_integer_real(F,I)),groups1270011288395367621r_real(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7462_member__le__sum,axiom,
    ! [I: complex,A: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I),A))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7463_member__le__sum,axiom,
    ! [I: int,A: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I),A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A,insert_int(I,bot_bot_set_int))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7464_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
       => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7465_member__le__sum,axiom,
    ! [I: real,A: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I),A))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A,insert_real(I,bot_bot_set_real))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7466_member__le__sum,axiom,
    ! [I: code_integer,A: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),A))
     => ( ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(A,insert_Code_integer(I,bot_bo3990330152332043303nteger))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X4))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,I)),groups6602215022474089585er_rat(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7467_member__le__sum,axiom,
    ! [I: complex,A: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I),A))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7468_member__le__sum,axiom,
    ! [I: int,A: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I),A))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A,insert_int(I,bot_bot_set_int))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A))) ) ) ) ).

% member_le_sum
tff(fact_7469_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_qk(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7470_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_ql(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7471_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7472_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(aTP_Lamm_pq(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7473_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_qm(F),set_or4665077453230672383an_nat(M,N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7474_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_qn(F),set_or4665077453230672383an_nat(M,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7475_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(aTP_Lamm_qo(F),set_or4665077453230672383an_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_7476_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_qm(F),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7477_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(aTP_Lamm_qn(F),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7478_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6591440286371151544t_real(aTP_Lamm_qo(F),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7479_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(N,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qp(G),N),M)),set_or4665077453230672383an_nat(N,M)) ).

% sum.atLeastLessThan_rev
tff(fact_7480_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(N,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_qq(G),N),M),set_or4665077453230672383an_nat(N,M)) ).

% sum.atLeastLessThan_rev
tff(fact_7481_sum_Onested__swap,axiom,
    ! [A2: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qr(A2)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qs(A2),N)),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% sum.nested_swap
tff(fact_7482_sum_Onested__swap,axiom,
    ! [A2: fun_nat_fun_nat_real,N: nat] : groups6591440286371151544t_real(aTP_Lamm_qt(A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qu(A2),N),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% sum.nested_swap
tff(fact_7483_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7484_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7485_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7486_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N))),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7487_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups2073611262835488442omplex(G,set_or4665077453230672383an_nat(M,N))),aa_nat_complex(G,N)) ) ) ) ).

% sum.head_if
tff(fact_7488_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% sum.head_if
tff(fact_7489_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N)) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or4665077453230672383an_nat(M,N))),aa_nat_int(G,N)) ) ) ) ).

% sum.head_if
tff(fact_7490_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N)) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(M,N))),aa_nat_nat(G,N)) ) ) ) ).

% sum.head_if
tff(fact_7491_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(M,N))),aa_nat_real(G,N)) ) ) ) ).

% sum.head_if
tff(fact_7492_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_7493_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or4665077453230672383an_nat(N,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(G),N),M)),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7494_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,set_or4665077453230672383an_nat(N,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_pz(G),N),M),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7495_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_qv(F),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(F,M)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_qv(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_7496_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_qw(F),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_qw(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_7497_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_qx(F),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_qx(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_7498_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups6591440286371151544t_real(aTP_Lamm_qy(F),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups6591440286371151544t_real(aTP_Lamm_qy(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_7499_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_qz(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_7500_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_ra(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_7501_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(aTP_Lamm_rb(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_7502_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer) = groups7501900531339628137nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,N))) ).

% mask_eq_sum_exp
tff(fact_7503_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,N))) ).

% mask_eq_sum_exp
tff(fact_7504_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,N))) ).

% mask_eq_sum_exp
tff(fact_7505_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_rc(G),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_7506_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_rd(G),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_7507_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_re(G)),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_7508_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups6591440286371151544t_real(aTP_Lamm_rf(G),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_7509_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,N))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7510_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X2)),groups7501900531339628137nteger(power_8256067586552552935nteger(X2),set_or1269000886237332187st_nat(M,N))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7511_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),M)),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7512_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X2)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),set_or1269000886237332187st_nat(M,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7513_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_7514_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,N))) ).

% mask_eq_sum_exp_nat
tff(fact_7515_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7516_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_7517_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_7518_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_7519_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7520_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_7521_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7522_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_7523_double__arith__series,axiom,
    ! [A2: complex,D2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_rg(A2),D2),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),D2))) ).

% double_arith_series
tff(fact_7524_double__arith__series,axiom,
    ! [A2: rat,D2: rat,N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rh(A2),D2),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A2)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D2))) ).

% double_arith_series
tff(fact_7525_double__arith__series,axiom,
    ! [A2: int,D2: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ri(A2),D2),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D2))) ).

% double_arith_series
tff(fact_7526_double__arith__series,axiom,
    ! [A2: nat,D2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rj(A2),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D2))) ).

% double_arith_series
tff(fact_7527_double__arith__series,axiom,
    ! [A2: real,D2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rk(A2),D2),set_or1269000886237332187st_nat(zero_zero_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D2))) ).

% double_arith_series
tff(fact_7528_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rl(A2),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_7529_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_7530_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7531_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7532_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7533_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7534_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7535_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7536_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7537_arith__series,axiom,
    ! [A2: int,D2: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ri(A2),D2),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7538_arith__series,axiom,
    ! [A2: nat,D2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rj(A2),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7539_sum__gp__offset,axiom,
    ! [X2: complex,M: nat,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X2 != one_one_complex )
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),M)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_nat(suc,N))))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)) ) ) ) ).

% sum_gp_offset
tff(fact_7540_sum__gp__offset,axiom,
    ! [X2: rat,M: nat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat) ) )
      & ( ( X2 != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),M)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ).

% sum_gp_offset
tff(fact_7541_sum__gp__offset,axiom,
    ! [X2: real,M: nat,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X2 != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ).

% sum_gp_offset
tff(fact_7542_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7543_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7544_Chebyshev__sum__upper,axiom,
    ! [N: nat,A2: fun_nat_rat,B2: fun_nat_rat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(A2,I2)),aa_nat_rat(A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(B2,J2)),aa_nat_rat(B2,I2))) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_rm(A2),B2),set_or4665077453230672383an_nat(zero_zero_nat,N)))),aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(A2,set_or4665077453230672383an_nat(zero_zero_nat,N))),groups2906978787729119204at_rat(B2,set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper
tff(fact_7545_Chebyshev__sum__upper,axiom,
    ! [N: nat,A2: fun_nat_int,B2: fun_nat_int] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(A2,I2)),aa_nat_int(A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(B2,J2)),aa_nat_int(B2,I2))) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_rn(A2),B2),set_or4665077453230672383an_nat(zero_zero_nat,N)))),aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(A2,set_or4665077453230672383an_nat(zero_zero_nat,N))),groups3539618377306564664at_int(B2,set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper
tff(fact_7546_Chebyshev__sum__upper,axiom,
    ! [N: nat,A2: fun_nat_real,B2: fun_nat_real] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,I2)),aa_nat_real(A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(B2,J2)),aa_nat_real(B2,I2))) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ro(A2),B2),set_or4665077453230672383an_nat(zero_zero_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(A2,set_or4665077453230672383an_nat(zero_zero_nat,N))),groups6591440286371151544t_real(B2,set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper
tff(fact_7547_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun_nat_nat,B2: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A2,I2)),aa_nat_nat(A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B2,J2)),aa_nat_nat(B2,I2))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rp(A2),B2)),set_or4665077453230672383an_nat(zero_zero_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A2),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(B2),set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_7548_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H)),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),N))),H)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rr(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7549_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),N)),aa_nat_rat(power_power_rat(Z),N))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rt(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7550_lemma__termdiff2,axiom,
    ! [H: real,Z: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rv(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7551_length__rule,axiom,
    ! [A2: array_bool,Xs: list_bool] : hoare_3067605981109127869le_nat(snga_assn_bool(A2,Xs),array_len_bool(A2),aa_lis1286645363786127574t_assn(aTP_Lamm_rw(A2),Xs)) ).

% length_rule
tff(fact_7552_length__rule,axiom,
    ! [A2: array_nat,Xs: list_nat] : hoare_3067605981109127869le_nat(snga_assn_nat(A2,Xs),array_len_nat(A2),aa_lis6041139530851742867t_assn(aTP_Lamm_rx(A2),Xs)) ).

% length_rule
tff(fact_7553_length__rule,axiom,
    ! [A2: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3067605981109127869le_nat(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A2),Xs),array_len_VEBT_VEBTi(A2),aa_lis7348694692495225848t_assn(aTP_Lamm_ry(A2),Xs)) ).

% length_rule
tff(fact_7554_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7555_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7556_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7557_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7558_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_lessThan_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),K)) ) ).

% lessThan_iff
tff(fact_7559_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_lessThan_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),K)) ) ).

% lessThan_iff
tff(fact_7560_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_7561_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).

% lessThan_iff
tff(fact_7562_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or5984915006950818249n_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_7563_lessThan__subset__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X2)),set_ord_lessThan_rat(Y3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% lessThan_subset_iff
tff(fact_7564_lessThan__subset__iff,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X2)),set_ord_lessThan_num(Y3)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3)) ) ).

% lessThan_subset_iff
tff(fact_7565_lessThan__subset__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X2)),aa_nat_set_nat(set_ord_lessThan_nat,Y3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ).

% lessThan_subset_iff
tff(fact_7566_lessThan__subset__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X2)),set_ord_lessThan_int(Y3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ).

% lessThan_subset_iff
tff(fact_7567_lessThan__subset__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X2)),set_or5984915006950818249n_real(Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% lessThan_subset_iff
tff(fact_7568_lessThan__minus__lessThan,axiom,
    ! [N: real,M: real] : minus_minus_set_real(set_or5984915006950818249n_real(N),set_or5984915006950818249n_real(M)) = set_or66887138388493659n_real(M,N) ).

% lessThan_minus_lessThan
tff(fact_7569_lessThan__minus__lessThan,axiom,
    ! [N: nat,M: nat] : minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),aa_nat_set_nat(set_ord_lessThan_nat,M)) = set_or4665077453230672383an_nat(M,N) ).

% lessThan_minus_lessThan
tff(fact_7570_lessThan__minus__lessThan,axiom,
    ! [N: int,M: int] : minus_minus_set_int(set_ord_lessThan_int(N),set_ord_lessThan_int(M)) = set_or4662586982721622107an_int(M,N) ).

% lessThan_minus_lessThan
tff(fact_7571_lessThan__minus__lessThan,axiom,
    ! [N: code_integer,M: code_integer] : minus_2355218937544613996nteger(set_or5754767410780653050nteger(N),set_or5754767410780653050nteger(M)) = set_or8404916559141939852nteger(M,N) ).

% lessThan_minus_lessThan
tff(fact_7572_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_7573_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_7574_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_7575_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_7576_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7577_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7578_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7579_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7580_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).

% sum.lessThan_Suc
tff(fact_7581_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% sum.lessThan_Suc
tff(fact_7582_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% sum.lessThan_Suc
tff(fact_7583_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% sum.lessThan_Suc
tff(fact_7584_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_7585_single__Diff__lessThan,axiom,
    ! [K: int] : minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_7586_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_7587_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,N: int] :
      ( ! [X4: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(Q,X4)),aa_int_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(N))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(N))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_rz(Q),P),set_ord_lessThan_int(N)) ) ) ).

% sum_diff_distrib
tff(fact_7588_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,N: real] :
      ( ! [X4: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X4)),aa_real_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(N))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_sa(Q),P),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_7589_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sb(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_7590_int__sum,axiom,
    ! [F: fun_int_nat,A: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A)) = groups4538972089207619220nt_int(aTP_Lamm_kb(F),A) ).

% int_sum
tff(fact_7591_int__sum,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups3539618377306564664at_int(aTP_Lamm_kc(F),A) ).

% int_sum
tff(fact_7592_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_sc,U)) ).

% lessThan_def
tff(fact_7593_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_sd,U)) ).

% lessThan_def
tff(fact_7594_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,U)) ).

% lessThan_def
tff(fact_7595_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_se,U)) ).

% lessThan_def
tff(fact_7596_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_sf,U)) ).

% lessThan_def
tff(fact_7597_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(N)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7598_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7599_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7600_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7601_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7602_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_7603_sum__subtractf__nat,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(G,X4)),aa_VEBT_VEBT_nat(F,X4))) )
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_sg(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A)),groups771621172384141258BT_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7604_sum__subtractf__nat,axiom,
    ! [A: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_sa(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A)),groups1935376822645274424al_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7605_sum__subtractf__nat,axiom,
    ! [A: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_rz(G),F),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(G,A)) ) ) ).

% sum_subtractf_nat
tff(fact_7606_sum__subtractf__nat,axiom,
    ! [A: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sb(G),F)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A)) ) ) ).

% sum_subtractf_nat
tff(fact_7607_sum__SucD,axiom,
    ! [F: fun_nat_nat,A: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_7608_sum__eq__Suc0__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( groups4541462559716669496nt_nat(F,A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A))
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7609_sum__eq__Suc0__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( groups7237345082560585321er_nat(F,A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
            & ( aa_Code_integer_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_Code_integer_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7610_sum__eq__Suc0__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( groups5693394587270226106ex_nat(F,A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A))
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7611_sum__eq__Suc0__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A))
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7612_sum__eq__1__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( groups4541462559716669496nt_nat(F,A) = one_one_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A))
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7613_sum__eq__1__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( groups7237345082560585321er_nat(F,A) = one_one_nat )
      <=> ? [X3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
            & ( aa_Code_integer_nat(F,X3) = one_one_nat )
            & ! [Xa4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_Code_integer_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7614_sum__eq__1__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( groups5693394587270226106ex_nat(F,A) = one_one_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A))
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7615_sum__eq__1__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A) = one_one_nat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A))
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7616_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sh(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_7617_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_si(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_7618_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_sj,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_hp(N),C2))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7619_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_sj,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_gk,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7620_sum__diff__nat,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
       => ( groups7237345082560585321er_nat(F,minus_2355218937544613996nteger(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups7237345082560585321er_nat(F,A)),groups7237345082560585321er_nat(F,B)) ) ) ) ).

% sum_diff_nat
tff(fact_7621_sum__diff__nat,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A)),groups5693394587270226106ex_nat(F,B)) ) ) ) ).

% sum_diff_nat
tff(fact_7622_sum__diff__nat,axiom,
    ! [B: set_int,A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(F,B)) ) ) ) ).

% sum_diff_nat
tff(fact_7623_sum__diff__nat,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B)) ) ) ) ).

% sum_diff_nat
tff(fact_7624_sum__diff1__nat,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
       => ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A)),aa_VEBT_VEBT_nat(F,A2)) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
       => ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = groups771621172384141258BT_nat(F,A) ) ) ) ).

% sum_diff1_nat
tff(fact_7625_sum__diff1__nat,axiom,
    ! [A2: real,A: set_real,F: fun_real_nat] :
      ( ( pp(aa_set_real_bool(member_real(A2),A))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A)),aa_real_nat(F,A2)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A2),A))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A) ) ) ) ).

% sum_diff1_nat
tff(fact_7626_sum__diff1__nat,axiom,
    ! [A2: int,A: set_int,F: fun_int_nat] :
      ( ( pp(aa_set_int_bool(member_int(A2),A))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A)),aa_int_nat(F,A2)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A2),A))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A) ) ) ) ).

% sum_diff1_nat
tff(fact_7627_sum__diff1__nat,axiom,
    ! [A2: nat,A: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A2),A))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A)),aa_nat_nat(F,A2)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A2),A))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A) ) ) ) ).

% sum_diff1_nat
tff(fact_7628_dbl__inc__def,axiom,
    ! [X2: real] : aa_real_real(neg_nu8295874005876285629c_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),X2)),one_one_real) ).

% dbl_inc_def
tff(fact_7629_dbl__inc__def,axiom,
    ! [X2: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(X2),X2)),one_one_rat) ).

% dbl_inc_def
tff(fact_7630_dbl__inc__def,axiom,
    ! [X2: int] : aa_int_int(neg_nu5851722552734809277nc_int,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),X2)),one_one_int) ).

% dbl_inc_def
tff(fact_7631_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_qk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7632_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7633_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7634_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7635_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_sk(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7636_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_sl(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7637_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_sm(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7638_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_qm(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7639_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_qn(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7640_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_qo(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7641_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,N: nat,R2: rat] : aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sn(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_7642_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,N: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_so(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_7643_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,N: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),R2)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sp(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_7644_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_7645_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = groups6591440286371151544t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_7646_sum_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sq(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% sum.nat_group
tff(fact_7647_sum_Onat__group,axiom,
    ! [G: fun_nat_real,K: nat,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sr(G),K),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% sum.nat_group
tff(fact_7648_one__diff__power__eq,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7649_one__diff__power__eq,axiom,
    ! [X2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X2)),groups7501900531339628137nteger(power_8256067586552552935nteger(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7650_one__diff__power__eq,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7651_one__diff__power__eq,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X2)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7652_one__diff__power__eq,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7653_power__diff__1__eq,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),one_one_complex) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),one_one_complex)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7654_power__diff__1__eq,axiom,
    ! [X2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),one_one_Code_integer)),groups7501900531339628137nteger(power_8256067586552552935nteger(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7655_power__diff__1__eq,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),N)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7656_power__diff__1__eq,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),one_one_int)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7657_power__diff__1__eq,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7658_geometric__sum,axiom,
    ! [X2: complex,N: nat] :
      ( ( X2 != one_one_complex )
     => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_7659_geometric__sum,axiom,
    ! [X2: rat,N: nat] :
      ( ( X2 != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),N)),one_one_rat)),aa_rat_rat(minus_minus_rat(X2),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_7660_geometric__sum,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 != one_one_real )
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)) ) ) ).

% geometric_sum
tff(fact_7661_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ss(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_st(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7662_lemma__termdiff1,axiom,
    ! [Z: code_integer,H: code_integer,M: nat] : groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_su(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_sv(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7663_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_sw(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_sx(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7664_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_sy(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_sz(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7665_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ta(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tb(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7666_sum__gp__strict,axiom,
    ! [X2: complex,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X2 != one_one_complex )
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)) ) ) ) ).

% sum_gp_strict
tff(fact_7667_sum__gp__strict,axiom,
    ! [X2: rat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X2 != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),N))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ).

% sum_gp_strict
tff(fact_7668_sum__gp__strict,axiom,
    ! [X2: real,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X2 != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ).

% sum_gp_strict
tff(fact_7669_power__diff__sumr2,axiom,
    ! [X2: complex,N: nat,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_tc(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7670_power__diff__sumr2,axiom,
    ! [X2: code_integer,N: nat,Y3: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),Y3)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_td(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7671_power__diff__sumr2,axiom,
    ! [X2: rat,N: nat,Y3: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),N)),aa_nat_rat(power_power_rat(Y3),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y3)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_te(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7672_power__diff__sumr2,axiom,
    ! [X2: int,N: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tf(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7673_power__diff__sumr2,axiom,
    ! [X2: real,N: nat,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tg(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7674_diff__power__eq__sum,axiom,
    ! [X2: complex,N: nat,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_nat(suc,N))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y3),aa_nat_nat(suc,N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_th(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7675_diff__power__eq__sum,axiom,
    ! [X2: code_integer,N: nat,Y3: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(Y3),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),Y3)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_ti(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7676_diff__power__eq__sum,axiom,
    ! [X2: rat,N: nat,Y3: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(Y3),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y3)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_tj(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7677_diff__power__eq__sum,axiom,
    ! [X2: int,N: nat,Y3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_nat(suc,N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y3),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tk(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7678_diff__power__eq__sum,axiom,
    ! [X2: real,N: nat,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tl(X2),N),Y3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7679_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_7680_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K6: rat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,P4)),K6)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7681_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K6: int,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K6)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7682_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K6: nat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K6)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7683_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K6: real,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K6)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K6))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7684_one__diff__power__eq_H,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_tm(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7685_one__diff__power__eq_H,axiom,
    ! [X2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X2),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X2)),groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_tn(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7686_one__diff__power__eq_H,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_to(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7687_one__diff__power__eq_H,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X2)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_tp(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7688_one__diff__power__eq_H,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tq(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7689_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_tr(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_ts(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),groups6591440286371151544t_real(aTP_Lamm_tt(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_7690_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_7691_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_7692_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_pq(F),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_7693_length__corresp,axiom,
    ! [Tree_array2: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBT] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array2)) = top_top_assn )
     => ( heap_Time_return_nat(size_s6755466524823107622T_VEBT(Tree_is)) = array_len_VEBT_VEBTi(Tree_array2) ) ) ).

% length_corresp
tff(fact_7694_length__corresp,axiom,
    ! [Tree_array2: array_VEBT_VEBTi,Tree_is: list_real] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array2)) = top_top_assn )
     => ( heap_Time_return_nat(size_size_list_real(Tree_is)) = array_len_VEBT_VEBTi(Tree_array2) ) ) ).

% length_corresp
tff(fact_7695_length__corresp,axiom,
    ! [Tree_array2: array_VEBT_VEBTi,Tree_is: list_bool] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array2)) = top_top_assn )
     => ( heap_Time_return_nat(size_size_list_bool(Tree_is)) = array_len_VEBT_VEBTi(Tree_array2) ) ) ).

% length_corresp
tff(fact_7696_length__corresp,axiom,
    ! [Tree_array2: array_VEBT_VEBTi,Tree_is: list_nat] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array2)) = top_top_assn )
     => ( heap_Time_return_nat(size_size_list_nat(Tree_is)) = array_len_VEBT_VEBTi(Tree_array2) ) ) ).

% length_corresp
tff(fact_7697_length__corresp,axiom,
    ! [Tree_array2: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi] :
      ( ( ex_ass463751140784270563_VEBTi(snga_assn_VEBT_VEBTi(Tree_array2)) = top_top_assn )
     => ( heap_Time_return_nat(size_s7982070591426661849_VEBTi(Tree_is)) = array_len_VEBT_VEBTi(Tree_array2) ) ) ).

% length_corresp
tff(fact_7698_sumr__cos__zero__one,axiom,
    ! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_tu,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_7699_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ? [K2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(S),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ).

% finite_nat_bounded
tff(fact_7700_finite__nat__iff__bounded,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(S),aa_nat_set_nat(set_ord_lessThan_nat,K3))) ) ).

% finite_nat_iff_bounded
tff(fact_7701_max__top,axiom,
    ! [X2: extended_enat] : ord_ma741700101516333627d_enat(top_to3028658606643905974d_enat,X2) = top_to3028658606643905974d_enat ).

% max_top
tff(fact_7702_max__top,axiom,
    ! [X2: set_Product_unit] : ord_ma508538259134630082t_unit(top_to1996260823553986621t_unit,X2) = top_to1996260823553986621t_unit ).

% max_top
tff(fact_7703_max__top,axiom,
    ! [X2: set_Numeral_num1] : ord_ma2202181864719522458l_num1(top_to3689904429138878997l_num1,X2) = top_to3689904429138878997l_num1 ).

% max_top
tff(fact_7704_max__top,axiom,
    ! [X2: set_real] : ord_max_set_real(top_top_set_real,X2) = top_top_set_real ).

% max_top
tff(fact_7705_max__top,axiom,
    ! [X2: set_bool] : ord_max_set_bool(top_top_set_bool,X2) = top_top_set_bool ).

% max_top
tff(fact_7706_max__top,axiom,
    ! [X2: set_nat] : ord_max_set_nat(top_top_set_nat,X2) = top_top_set_nat ).

% max_top
tff(fact_7707_max__top2,axiom,
    ! [X2: extended_enat] : ord_ma741700101516333627d_enat(X2,top_to3028658606643905974d_enat) = top_to3028658606643905974d_enat ).

% max_top2
tff(fact_7708_max__top2,axiom,
    ! [X2: set_Product_unit] : ord_ma508538259134630082t_unit(X2,top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).

% max_top2
tff(fact_7709_max__top2,axiom,
    ! [X2: set_Numeral_num1] : ord_ma2202181864719522458l_num1(X2,top_to3689904429138878997l_num1) = top_to3689904429138878997l_num1 ).

% max_top2
tff(fact_7710_max__top2,axiom,
    ! [X2: set_real] : ord_max_set_real(X2,top_top_set_real) = top_top_set_real ).

% max_top2
tff(fact_7711_max__top2,axiom,
    ! [X2: set_bool] : ord_max_set_bool(X2,top_top_set_bool) = top_top_set_bool ).

% max_top2
tff(fact_7712_max__top2,axiom,
    ! [X2: set_nat] : ord_max_set_nat(X2,top_top_set_nat) = top_top_set_nat ).

% max_top2
tff(fact_7713_merge__true__star,axiom,
    aa_assn_assn(times_times_assn(top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_7714_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_7715_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_7716_top_Oextremum__strict,axiom,
    ! [A2: set_Product_unit] : ~ ord_le8056459307392131481t_unit(top_to1996260823553986621t_unit,A2) ).

% top.extremum_strict
tff(fact_7717_top_Oextremum__strict,axiom,
    ! [A2: set_Numeral_num1] : ~ ord_le526730876122248049l_num1(top_to3689904429138878997l_num1,A2) ).

% top.extremum_strict
tff(fact_7718_top_Oextremum__strict,axiom,
    ! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(top_top_set_real),A2)) ).

% top.extremum_strict
tff(fact_7719_top_Oextremum__strict,axiom,
    ! [A2: set_bool] : ~ ord_less_set_bool(top_top_set_bool,A2) ).

% top.extremum_strict
tff(fact_7720_top_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(top_top_set_nat),A2)) ).

% top.extremum_strict
tff(fact_7721_top_Onot__eq__extremum,axiom,
    ! [A2: set_Product_unit] :
      ( ( A2 != top_to1996260823553986621t_unit )
    <=> ord_le8056459307392131481t_unit(A2,top_to1996260823553986621t_unit) ) ).

% top.not_eq_extremum
tff(fact_7722_top_Onot__eq__extremum,axiom,
    ! [A2: set_Numeral_num1] :
      ( ( A2 != top_to3689904429138878997l_num1 )
    <=> ord_le526730876122248049l_num1(A2,top_to3689904429138878997l_num1) ) ).

% top.not_eq_extremum
tff(fact_7723_top_Onot__eq__extremum,axiom,
    ! [A2: set_real] :
      ( ( A2 != top_top_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(A2),top_top_set_real)) ) ).

% top.not_eq_extremum
tff(fact_7724_top_Onot__eq__extremum,axiom,
    ! [A2: set_bool] :
      ( ( A2 != top_top_set_bool )
    <=> ord_less_set_bool(A2,top_top_set_bool) ) ).

% top.not_eq_extremum
tff(fact_7725_top_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != top_top_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_set_nat(A2),top_top_set_nat)) ) ).

% top.not_eq_extremum
tff(fact_7726_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(top_top_assn),aa_assn_assn(times_times_assn(top_top_assn),P)) = aa_assn_assn(times_times_assn(top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_7727_top__option__def,axiom,
    top_to7662761140297458691t_unit = some_s8043159101241848530t_unit(top_to1996260823553986621t_unit) ).

% top_option_def
tff(fact_7728_top__option__def,axiom,
    top_to176924430543178203l_num1 = some_s513430669971965098l_num1(top_to3689904429138878997l_num1) ).

% top_option_def
tff(fact_7729_top__option__def,axiom,
    top_to1083748111577038690t_real = some_set_real(top_top_set_real) ).

% top_option_def
tff(fact_7730_top__option__def,axiom,
    top_to1856094199974014989t_bool = some_set_bool(top_top_set_bool) ).

% top_option_def
tff(fact_7731_top__option__def,axiom,
    top_to4826455019444611206et_nat = some_set_nat(top_top_set_nat) ).

% top_option_def
tff(fact_7732_top__greatest,axiom,
    ! [A2: set_Product_unit] : ord_le3507040750410214029t_unit(A2,top_to1996260823553986621t_unit) ).

% top_greatest
tff(fact_7733_top__greatest,axiom,
    ! [A2: set_Numeral_num1] : ord_le5200684355995106405l_num1(A2,top_to3689904429138878997l_num1) ).

% top_greatest
tff(fact_7734_top__greatest,axiom,
    ! [A2: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(A2),top_top_set_real)) ).

% top_greatest
tff(fact_7735_top__greatest,axiom,
    ! [A2: set_bool] : ord_less_eq_set_bool(A2,top_top_set_bool) ).

% top_greatest
tff(fact_7736_top__greatest,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),top_top_set_nat)) ).

% top_greatest
tff(fact_7737_top__greatest,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),top_top_set_int)) ).

% top_greatest
tff(fact_7738_top_Oextremum__unique,axiom,
    ! [A2: set_Product_unit] :
      ( ord_le3507040750410214029t_unit(top_to1996260823553986621t_unit,A2)
    <=> ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_unique
tff(fact_7739_top_Oextremum__unique,axiom,
    ! [A2: set_Numeral_num1] :
      ( ord_le5200684355995106405l_num1(top_to3689904429138878997l_num1,A2)
    <=> ( A2 = top_to3689904429138878997l_num1 ) ) ).

% top.extremum_unique
tff(fact_7740_top_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(top_top_set_real),A2))
    <=> ( A2 = top_top_set_real ) ) ).

% top.extremum_unique
tff(fact_7741_top_Oextremum__unique,axiom,
    ! [A2: set_bool] :
      ( ord_less_eq_set_bool(top_top_set_bool,A2)
    <=> ( A2 = top_top_set_bool ) ) ).

% top.extremum_unique
tff(fact_7742_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(top_top_set_nat),A2))
    <=> ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
tff(fact_7743_top_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(top_top_set_int),A2))
    <=> ( A2 = top_top_set_int ) ) ).

% top.extremum_unique
tff(fact_7744_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_unit] :
      ( ord_le3507040750410214029t_unit(top_to1996260823553986621t_unit,A2)
     => ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_uniqueI
tff(fact_7745_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Numeral_num1] :
      ( ord_le5200684355995106405l_num1(top_to3689904429138878997l_num1,A2)
     => ( A2 = top_to3689904429138878997l_num1 ) ) ).

% top.extremum_uniqueI
tff(fact_7746_top_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(top_top_set_real),A2))
     => ( A2 = top_top_set_real ) ) ).

% top.extremum_uniqueI
tff(fact_7747_top_Oextremum__uniqueI,axiom,
    ! [A2: set_bool] :
      ( ord_less_eq_set_bool(top_top_set_bool,A2)
     => ( A2 = top_top_set_bool ) ) ).

% top.extremum_uniqueI
tff(fact_7748_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(top_top_set_nat),A2))
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
tff(fact_7749_top_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(top_top_set_int),A2))
     => ( A2 = top_top_set_int ) ) ).

% top.extremum_uniqueI
tff(fact_7750_ent__true,axiom,
    ! [P: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),top_top_assn)) ).

% ent_true
tff(fact_7751_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(Q),top_top_assn))) ) ).

% ent_true_drop(2)
tff(fact_7752_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(Q),top_top_assn)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(P),R)),aa_assn_assn(times_times_assn(Q),top_top_assn))) ) ).

% ent_true_drop(1)
tff(fact_7753_ent__refl__true,axiom,
    ! [A: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(times_times_assn(A),top_top_assn))) ).

% ent_refl_true
tff(fact_7754_ent__star__mono__true,axiom,
    ! [A: assn,A6: assn,B: assn,B7: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(times_times_assn(A6),top_top_assn)))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),aa_assn_assn(times_times_assn(B7),top_top_assn)))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),top_top_assn)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A6),B7)),top_top_assn))) ) ) ).

% ent_star_mono_true
tff(fact_7755_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn)),H))
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H5)) ) ).

% mod_star_trueE
tff(fact_7756_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(rep_assn(P),H))
     => pp(aa_Pro2991721470636411351t_bool(rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn)),H)) ) ).

% mod_star_trueI
tff(fact_7757_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M5))
         => ? [N8: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N8))
              & pp(aa_set_nat_bool(member_nat(N8),S)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S)) ) ).

% unbounded_k_infinite
tff(fact_7758_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ! [M7: nat] :
        ? [N7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N7))
          & pp(aa_set_nat_bool(member_nat(N7),S)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_7759_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ! [M7: nat] :
        ? [N7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N7))
          & pp(aa_set_nat_bool(member_nat(N7),S)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_7760_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_tv,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_7761_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_tw,Z)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_7762_vebt__assn__raw_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y3: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,X2),Xa) = Y3 )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(X2,Xa))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ! [Ai2: bool,Bi2: bool] :
                  ( ( Xa = vEBT_Leafi(Ai2,Bi2) )
                 => ( ( Y3 = aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_bool_bool(fequal_bool(Ai2),A3)),aa_bool_bool(fequal_bool(Bi2),B3))) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3),vEBT_Leafi(Ai2,Bi2))) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                   => ( ( Y3 = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,Mmoi),Mmo)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Degi),Deg2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_ee(Tree_list),Tree_array))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va3,Vb3),Vc3) )
                 => ! [Vd: bool,Ve: bool] :
                      ( ( Xa = vEBT_Leafi(Vd,Ve) )
                     => ( ( Y3 = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va3,Vb3),Vc3),vEBT_Leafi(Vd,Ve))) ) ) )
             => ~ ! [Vd: bool,Ve: bool] :
                    ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Vd),Ve) )
                   => ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va3,Vb3),Vc3) )
                       => ( ( Y3 = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_bool_VEBT_VEBT(vEBT_Leaf(Vd),Ve),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va3,Vb3),Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_7763_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_bool_VEBT_VEBT(vEBT_Leaf(A2),B2)) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_7764_pochhammer__double,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,Z),N))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_7765_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_7766_pochhammer__double,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_7767_finite__option__UNIV,axiom,
    ( finite1345302120164226195on_int(top_to6430115241214627170on_int)
  <=> pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ) ).

% finite_option_UNIV
tff(fact_7768_finite__option__UNIV,axiom,
    ( finite6785661671136154180nteger(top_to5929521628599800467nteger)
  <=> pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,top_to4645266643341252675nteger)) ) ).

% finite_option_UNIV
tff(fact_7769_finite__option__UNIV,axiom,
    ( finite2569390945932476949omplex(top_to6180147692022559204omplex)
  <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,top_top_set_complex)) ) ).

% finite_option_UNIV
tff(fact_7770_finite__option__UNIV,axiom,
    ( finite1445617369574913404t_unit(top_to2690860209552263555t_unit)
  <=> finite4290736615968046902t_unit(top_to1996260823553986621t_unit) ) ).

% finite_option_UNIV
tff(fact_7771_finite__option__UNIV,axiom,
    ( finite3139260975159805780l_num1(top_to4428395536652758875l_num1)
  <=> finite1111429032697314574l_num1(top_to3689904429138878997l_num1) ) ).

% finite_option_UNIV
tff(fact_7772_finite__option__UNIV,axiom,
    ( finite5731380839103853331n_real(top_to853713521313446370n_real)
  <=> pp(aa_set_real_bool(finite_finite_real,top_top_set_real)) ) ).

% finite_option_UNIV
tff(fact_7773_finite__option__UNIV,axiom,
    ( finite3444297449656966n_bool(top_to8850353440797054349n_bool)
  <=> finite_finite_bool(top_top_set_bool) ) ).

% finite_option_UNIV
tff(fact_7774_finite__option__UNIV,axiom,
    ( finite5523153139673422903on_nat(top_to8920198386146353926on_nat)
  <=> pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ) ).

% finite_option_UNIV
tff(fact_7775_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_complex(aTP_Lamm_tx(P)) = top_top_set_complex ) )
      & ( ~ pp(P)
       => ( collect_complex(aTP_Lamm_tx(P)) = bot_bot_set_complex ) ) ) ).

% Collect_const
tff(fact_7776_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collec213857154873943460nt_int(aTP_Lamm_ty(P)) = top_to4366644338036079209nt_int ) )
      & ( ~ pp(P)
       => ( collec213857154873943460nt_int(aTP_Lamm_ty(P)) = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_const
tff(fact_7777_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_int(aTP_Lamm_tz(P)) = top_top_set_int ) )
      & ( ~ pp(P)
       => ( collect_int(aTP_Lamm_tz(P)) = bot_bot_set_int ) ) ) ).

% Collect_const
tff(fact_7778_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_Product_unit(aTP_Lamm_ua(P)) = top_to1996260823553986621t_unit ) )
      & ( ~ pp(P)
       => ( collect_Product_unit(aTP_Lamm_ua(P)) = bot_bo3957492148770167129t_unit ) ) ) ).

% Collect_const
tff(fact_7779_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_Numeral_num1(aTP_Lamm_ub(P)) = top_to3689904429138878997l_num1 ) )
      & ( ~ pp(P)
       => ( collect_Numeral_num1(aTP_Lamm_ub(P)) = bot_bo5651135754355059505l_num1 ) ) ) ).

% Collect_const
tff(fact_7780_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_real(aTP_Lamm_uc(P)) = top_top_set_real ) )
      & ( ~ pp(P)
       => ( collect_real(aTP_Lamm_uc(P)) = bot_bot_set_real ) ) ) ).

% Collect_const
tff(fact_7781_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_bool(aTP_Lamm_ud(P)) = top_top_set_bool ) )
      & ( ~ pp(P)
       => ( collect_bool(aTP_Lamm_ud(P)) = bot_bot_set_bool ) ) ) ).

% Collect_const
tff(fact_7782_Collect__const,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( collect_nat(aa_bool_fun_nat_bool(aTP_Lamm_ue,P)) = top_top_set_nat ) )
      & ( ~ pp(P)
       => ( collect_nat(aa_bool_fun_nat_bool(aTP_Lamm_ue,P)) = bot_bot_set_nat ) ) ) ).

% Collect_const
tff(fact_7783_finite__Collect__not,axiom,
    ! [P: fun_Pr803113274115125473t_bool] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_uf(P)))
      <=> finite2998713641127702882nt_int(top_to4366644338036079209nt_int) ) ) ).

% finite_Collect_not
tff(fact_7784_finite__Collect__not,axiom,
    ! [P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_ug(P))))
      <=> pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ) ) ).

% finite_Collect_not
tff(fact_7785_finite__Collect__not,axiom,
    ! [P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_uh(P))))
      <=> pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,top_to4645266643341252675nteger)) ) ) ).

% finite_Collect_not
tff(fact_7786_finite__Collect__not,axiom,
    ! [P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ui(P))))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,top_top_set_complex)) ) ) ).

% finite_Collect_not
tff(fact_7787_finite__Collect__not,axiom,
    ! [P: fun_Pr1572506220757514509t_bool] :
      ( finite4290736615968046902t_unit(collect_Product_unit(P))
     => ( finite4290736615968046902t_unit(collect_Product_unit(aTP_Lamm_uj(P)))
      <=> finite4290736615968046902t_unit(top_to1996260823553986621t_unit) ) ) ).

% finite_Collect_not
tff(fact_7788_finite__Collect__not,axiom,
    ! [P: fun_Nu28140169051909368851_bool] :
      ( finite1111429032697314574l_num1(collect_Numeral_num1(P))
     => ( finite1111429032697314574l_num1(collect_Numeral_num1(aTP_Lamm_uk(P)))
      <=> finite1111429032697314574l_num1(top_to3689904429138878997l_num1) ) ) ).

% finite_Collect_not
tff(fact_7789_finite__Collect__not,axiom,
    ! [P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aTP_Lamm_ul(P))))
      <=> pp(aa_set_real_bool(finite_finite_real,top_top_set_real)) ) ) ).

% finite_Collect_not
tff(fact_7790_finite__Collect__not,axiom,
    ! [P: fun_bool_bool] :
      ( finite_finite_bool(collect_bool(P))
     => ( finite_finite_bool(collect_bool(aTP_Lamm_um(P)))
      <=> finite_finite_bool(top_top_set_bool) ) ) ).

% finite_Collect_not
tff(fact_7791_finite__Collect__not,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_un(P))))
      <=> pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ) ) ).

% finite_Collect_not
tff(fact_7792_Diff__UNIV,axiom,
    ! [A: set_int] : minus_minus_set_int(A,top_top_set_int) = bot_bot_set_int ).

% Diff_UNIV
tff(fact_7793_Diff__UNIV,axiom,
    ! [A: set_Product_unit] : minus_6452836326544984404t_unit(A,top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).

% Diff_UNIV
tff(fact_7794_Diff__UNIV,axiom,
    ! [A: set_Numeral_num1] : minus_8146479932129876780l_num1(A,top_to3689904429138878997l_num1) = bot_bo5651135754355059505l_num1 ).

% Diff_UNIV
tff(fact_7795_Diff__UNIV,axiom,
    ! [A: set_real] : minus_minus_set_real(A,top_top_set_real) = bot_bot_set_real ).

% Diff_UNIV
tff(fact_7796_Diff__UNIV,axiom,
    ! [A: set_bool] : minus_minus_set_bool(A,top_top_set_bool) = bot_bot_set_bool ).

% Diff_UNIV
tff(fact_7797_Diff__UNIV,axiom,
    ! [A: set_nat] : minus_minus_set_nat(A,top_top_set_nat) = bot_bot_set_nat ).

% Diff_UNIV
tff(fact_7798_pochhammer__0,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_7799_pochhammer__0,axiom,
    ! [A2: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_7800_pochhammer__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_7801_pochhammer__0,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_7802_powser__sums__zero__iff,axiom,
    ! [A2: fun_nat_complex,X2: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_le(A2)),X2))
    <=> ( aa_nat_complex(A2,zero_zero_nat) = X2 ) ) ).

% powser_sums_zero_iff
tff(fact_7803_powser__sums__zero__iff,axiom,
    ! [A2: fun_nat_real,X2: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_lg(A2)),X2))
    <=> ( aa_nat_real(A2,zero_zero_nat) = X2 ) ) ).

% powser_sums_zero_iff
tff(fact_7804_subset__UNIV,axiom,
    ! [A: set_Product_unit] : ord_le3507040750410214029t_unit(A,top_to1996260823553986621t_unit) ).

% subset_UNIV
tff(fact_7805_subset__UNIV,axiom,
    ! [A: set_Numeral_num1] : ord_le5200684355995106405l_num1(A,top_to3689904429138878997l_num1) ).

% subset_UNIV
tff(fact_7806_subset__UNIV,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(A),top_top_set_real)) ).

% subset_UNIV
tff(fact_7807_subset__UNIV,axiom,
    ! [A: set_bool] : ord_less_eq_set_bool(A,top_top_set_bool) ).

% subset_UNIV
tff(fact_7808_subset__UNIV,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),top_top_set_nat)) ).

% subset_UNIV
tff(fact_7809_subset__UNIV,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),top_top_set_int)) ).

% subset_UNIV
tff(fact_7810_pochhammer__pos,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X2),N))) ) ).

% pochhammer_pos
tff(fact_7811_pochhammer__pos,axiom,
    ! [X2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X2),N))) ) ).

% pochhammer_pos
tff(fact_7812_pochhammer__pos,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X2),N))) ) ).

% pochhammer_pos
tff(fact_7813_pochhammer__pos,axiom,
    ! [X2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,X2),N))) ) ).

% pochhammer_pos
tff(fact_7814_pochhammer__eq__0__mono,axiom,
    ! [A2: complex,N: nat,M: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A2),N) = zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A2),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7815_pochhammer__eq__0__mono,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7816_pochhammer__eq__0__mono,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A2),N) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A2),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7817_pochhammer__neq__0__mono,axiom,
    ! [A2: complex,M: nat,N: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A2),M) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A2),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7818_pochhammer__neq__0__mono,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),M) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7819_pochhammer__neq__0__mono,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A2),M) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A2),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7820_not__UNIV__le__Icc,axiom,
    ! [L: nat,H: nat] : ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(top_top_set_nat),set_or1269000886237332187st_nat(L,H))) ).

% not_UNIV_le_Icc
tff(fact_7821_not__UNIV__le__Icc,axiom,
    ! [L: int,H: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(top_top_set_int),set_or1266510415728281911st_int(L,H))) ).

% not_UNIV_le_Icc
tff(fact_7822_not__UNIV__le__Icc,axiom,
    ! [L: real,H: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(top_top_set_real),set_or1222579329274155063t_real(L,H))) ).

% not_UNIV_le_Icc
tff(fact_7823_pochhammer__nonneg,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7824_pochhammer__nonneg,axiom,
    ! [X2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7825_pochhammer__nonneg,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7826_pochhammer__nonneg,axiom,
    ! [X2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7827_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_7828_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_7829_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_7830_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_7831_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_7832_powser__sums__zero,axiom,
    ! [A2: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_le(A2)),aa_nat_complex(A2,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7833_powser__sums__zero,axiom,
    ! [A2: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_lg(A2)),aa_nat_real(A2,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7834_pochhammer__rec,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_7835_pochhammer__rec,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_7836_pochhammer__rec,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),N)) ).

% pochhammer_rec
tff(fact_7837_pochhammer__rec,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),N)) ).

% pochhammer_rec
tff(fact_7838_pochhammer__Suc,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A2),N)),aa_rat_rat(plus_plus_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_7839_pochhammer__Suc,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_7840_pochhammer__Suc,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_7841_pochhammer__Suc,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_7842_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_7843_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)) ).

% pochhammer_rec'
tff(fact_7844_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N)) ).

% pochhammer_rec'
tff(fact_7845_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_7846_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).

% pochhammer_product'
tff(fact_7847_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_7848_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_7849_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_7850_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7851_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7852_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),M)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7853_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7854_VEBT__internal_Oheight_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A3: bool,B3: bool] : X2 != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) ) ).

% VEBT_internal.height.cases
tff(fact_7855_sums__if_H,axiom,
    ! [G: fun_nat_real,X2: real] :
      ( pp(aa_real_bool(sums_real(G),X2))
     => pp(aa_real_bool(sums_real(aTP_Lamm_uo(G)),X2)) ) ).

% sums_if'
tff(fact_7856_sums__if,axiom,
    ! [G: fun_nat_real,X2: real,F: fun_nat_real,Y3: real] :
      ( pp(aa_real_bool(sums_real(G),X2))
     => ( pp(aa_real_bool(sums_real(F),Y3))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_up(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))) ) ) ).

% sums_if
tff(fact_7857_geometric__sums,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C2)),one_one_real))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(power_power_real,C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C2)))) ) ).

% geometric_sums
tff(fact_7858_geometric__sums,axiom,
    ! [C2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C2)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(power_power_complex,C2)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),C2)))) ) ).

% geometric_sums
tff(fact_7859_sums__group,axiom,
    ! [F: fun_nat_nat,S2: nat,K: nat] :
      ( pp(aa_nat_bool(sums_nat(F),S2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(sums_nat(aa_nat_fun_nat_nat(aTP_Lamm_sq(F),K)),S2)) ) ) ).

% sums_group
tff(fact_7860_sums__group,axiom,
    ! [F: fun_nat_real,S2: real,K: nat] :
      ( pp(aa_real_bool(sums_real(F),S2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_sr(F),K)),S2)) ) ) ).

% sums_group
tff(fact_7861_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_uq),one_one_real)) ).

% power_half_series
tff(fact_7862_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_ur),zero_zero_complex)) ).

% sums_zero
tff(fact_7863_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_jz),zero_zero_real)) ).

% sums_zero
tff(fact_7864_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_bj),zero_zero_nat)) ).

% sums_zero
tff(fact_7865_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_us),zero_zero_int)) ).

% sums_zero
tff(fact_7866_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S: real,A: set_nat,S6: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A))
       => ( ( S6 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ut(G),F),A)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_uu(G),A),F)),S6)) ) ) ) ).

% sums_If_finite_set'
tff(fact_7867_UNIV__def,axiom,
    top_top_set_int = collect_int(aTP_Lamm_uv) ).

% UNIV_def
tff(fact_7868_UNIV__def,axiom,
    top_top_set_complex = collect_complex(aTP_Lamm_uw) ).

% UNIV_def
tff(fact_7869_UNIV__def,axiom,
    top_to4366644338036079209nt_int = collec213857154873943460nt_int(aTP_Lamm_ux) ).

% UNIV_def
tff(fact_7870_UNIV__def,axiom,
    top_to1996260823553986621t_unit = collect_Product_unit(aTP_Lamm_uy) ).

% UNIV_def
tff(fact_7871_UNIV__def,axiom,
    top_to3689904429138878997l_num1 = collect_Numeral_num1(aTP_Lamm_uz) ).

% UNIV_def
tff(fact_7872_UNIV__def,axiom,
    top_top_set_real = collect_real(aTP_Lamm_va) ).

% UNIV_def
tff(fact_7873_UNIV__def,axiom,
    top_top_set_bool = collect_bool(aTP_Lamm_vb) ).

% UNIV_def
tff(fact_7874_UNIV__def,axiom,
    top_top_set_nat = collect_nat(aTP_Lamm_vc) ).

% UNIV_def
tff(fact_7875_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S2: real,T: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(G,N3)))
     => ( pp(aa_real_bool(sums_real(F),S2))
       => ( pp(aa_real_bool(sums_real(G),T))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S2),T)) ) ) ) ).

% sums_le
tff(fact_7876_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S2: nat,T: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(G,N3)))
     => ( pp(aa_nat_bool(sums_nat(F),S2))
       => ( pp(aa_nat_bool(sums_nat(G),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S2),T)) ) ) ) ).

% sums_le
tff(fact_7877_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S2: int,T: int] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(G,N3)))
     => ( pp(aa_int_bool(sums_int(F),S2))
       => ( pp(aa_int_bool(sums_int(G),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S2),T)) ) ) ) ).

% sums_le
tff(fact_7878_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kw(I),F)),aa_nat_complex(F,I))) ).

% sums_single
tff(fact_7879_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_vd(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_7880_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ve(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_7881_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_vf(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_7882_sums__add,axiom,
    ! [F: fun_nat_real,A2: real,G: fun_nat_real,B2: real] :
      ( pp(aa_real_bool(sums_real(F),A2))
     => ( pp(aa_real_bool(sums_real(G),B2))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oq(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))) ) ) ).

% sums_add
tff(fact_7883_sums__add,axiom,
    ! [F: fun_nat_nat,A2: nat,G: fun_nat_nat,B2: nat] :
      ( pp(aa_nat_bool(sums_nat(F),A2))
     => ( pp(aa_nat_bool(sums_nat(G),B2))
       => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oo(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% sums_add
tff(fact_7884_sums__add,axiom,
    ! [F: fun_nat_int,A2: int,G: fun_nat_int,B2: int] :
      ( pp(aa_int_bool(sums_int(F),A2))
     => ( pp(aa_int_bool(sums_int(G),B2))
       => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_vg(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% sums_add
tff(fact_7885_sums__mult,axiom,
    ! [F: fun_nat_real,A2: real,C2: real] :
      ( pp(aa_real_bool(sums_real(F),A2))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_vh(F),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2))) ) ).

% sums_mult
tff(fact_7886_sums__mult2,axiom,
    ! [F: fun_nat_real,A2: real,C2: real] :
      ( pp(aa_real_bool(sums_real(F),A2))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_nn(F),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))) ) ).

% sums_mult2
tff(fact_7887_sums__diff,axiom,
    ! [F: fun_nat_real,A2: real,G: fun_nat_real,B2: real] :
      ( pp(aa_real_bool(sums_real(F),A2))
     => ( pp(aa_real_bool(sums_real(G),B2))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_nf(F),G)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))) ) ) ).

% sums_diff
tff(fact_7888_sums__divide,axiom,
    ! [F: fun_nat_complex,A2: complex,C2: complex] :
      ( pp(aa_complex_bool(sums_complex(F),A2))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_vi(F),C2)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2))) ) ).

% sums_divide
tff(fact_7889_sums__divide,axiom,
    ! [F: fun_nat_real,A2: real,C2: real] :
      ( pp(aa_real_bool(sums_real(F),A2))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ok(F),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2))) ) ).

% sums_divide
tff(fact_7890_sums__sum,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VE2989161857680785728t_real,X2: fun_VEBT_VEBT_real] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
         => pp(aa_real_bool(sums_real(aa_VEB3736018289492012273t_real(F,I2)),aa_VEBT_VEBT_real(X2,I2))) )
     => pp(aa_real_bool(sums_real(aa_fun6147343916763992189t_real(aTP_Lamm_vk(I6),F)),groups2240296850493347238T_real(X2,I6))) ) ).

% sums_sum
tff(fact_7891_sums__sum,axiom,
    ! [I6: set_real,F: fun_re4238438383274954274t_real,X2: fun_real_real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I6))
         => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(F,I2)),aa_real_real(X2,I2))) )
     => pp(aa_real_bool(sums_real(aa_fun628238740774635481t_real(aTP_Lamm_vm(I6),F)),groups8097168146408367636l_real(X2,I6))) ) ).

% sums_sum
tff(fact_7892_sums__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_real,X2: fun_int_real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I6))
         => pp(aa_real_bool(sums_real(aa_int_fun_nat_real(F,I2)),aa_int_real(X2,I2))) )
     => pp(aa_real_bool(sums_real(aa_fun2842512154077127769t_real(aTP_Lamm_vo(I6),F)),groups8778361861064173332t_real(X2,I6))) ) ).

% sums_sum
tff(fact_7893_sums__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_nat,X2: fun_nat_nat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => pp(aa_nat_bool(sums_nat(aa_nat_fun_nat_nat(F,I2)),aa_nat_nat(X2,I2))) )
     => pp(aa_nat_bool(sums_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_vp(I6),F)),aa_set_nat_nat(groups3542108847815614940at_nat(X2),I6))) ) ).

% sums_sum
tff(fact_7894_sums__sum,axiom,
    ! [I6: set_complex,F: fun_co3258106519541737638omplex,X2: fun_complex_complex] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),I6))
         => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(F,I2)),aa_complex_complex(X2,I2))) )
     => pp(aa_complex_bool(sums_complex(aa_fun5003373910569599327omplex(aTP_Lamm_vr(I6),F)),groups7754918857620584856omplex(X2,I6))) ) ).

% sums_sum
tff(fact_7895_sums__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_real,X2: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(F,I2)),aa_nat_real(X2,I2))) )
     => pp(aa_real_bool(sums_real(aa_fun7039663905532832253t_real(aTP_Lamm_vs(I6),F)),groups6591440286371151544t_real(X2,I6))) ) ).

% sums_sum
tff(fact_7896_sums__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_int,X2: fun_int_int] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I6))
         => pp(aa_int_bool(sums_int(aa_int_fun_nat_int(F,I2)),aa_int_int(X2,I2))) )
     => pp(aa_int_bool(sums_int(aa_fun8710705519803144921at_int(aTP_Lamm_vu(I6),F)),groups4538972089207619220nt_int(X2,I6))) ) ).

% sums_sum
tff(fact_7897_sums__mult__iff,axiom,
    ! [C2: complex,F: fun_nat_complex,D2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vv(C2),F)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C2),D2)))
      <=> pp(aa_complex_bool(sums_complex(F),D2)) ) ) ).

% sums_mult_iff
tff(fact_7898_sums__mult__iff,axiom,
    ! [C2: real,F: fun_nat_real,D2: real] :
      ( ( C2 != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_nj(C2),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),D2)))
      <=> pp(aa_real_bool(sums_real(F),D2)) ) ) ).

% sums_mult_iff
tff(fact_7899_sums__mult2__iff,axiom,
    ! [C2: complex,F: fun_nat_complex,D2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vw(C2),F)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,D2),C2)))
      <=> pp(aa_complex_bool(sums_complex(F),D2)) ) ) ).

% sums_mult2_iff
tff(fact_7900_sums__mult2__iff,axiom,
    ! [C2: real,F: fun_nat_real,D2: real] :
      ( ( C2 != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_vx(C2),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D2),C2)))
      <=> pp(aa_real_bool(sums_real(F),D2)) ) ) ).

% sums_mult2_iff
tff(fact_7901_sums__mult__D,axiom,
    ! [C2: complex,F: fun_nat_complex,A2: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vv(C2),F)),A2))
     => ( ( C2 != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),C2))) ) ) ).

% sums_mult_D
tff(fact_7902_sums__mult__D,axiom,
    ! [C2: real,F: fun_nat_real,A2: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_nj(C2),F)),A2))
     => ( ( C2 != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2))) ) ) ).

% sums_mult_D
tff(fact_7903_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S2: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aTP_Lamm_vy(F)),S2))
       => pp(aa_complex_bool(sums_complex(F),S2)) ) ) ).

% sums_Suc_imp
tff(fact_7904_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_pq(F)),S2))
       => pp(aa_real_bool(sums_real(F),S2)) ) ) ).

% sums_Suc_imp
tff(fact_7905_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_pq(F)),S2))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_7906_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_pq(F)),L))
     => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_7907_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_pp(F)),L))
     => pp(aa_nat_bool(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_7908_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_ql(F)),L))
     => pp(aa_int_bool(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_7909_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_complex,S2: complex] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ( aa_nat_complex(F,I2) = zero_zero_complex ) )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vz(N),F)),S2))
      <=> pp(aa_complex_bool(sums_complex(F),S2)) ) ) ).

% sums_zero_iff_shift
tff(fact_7910_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S2: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_wa(N),F)),S2))
      <=> pp(aa_real_bool(sums_real(F),S2)) ) ) ).

% sums_zero_iff_shift
tff(fact_7911_sums__finite,axiom,
    ! [N6: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N6))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N6))
           => ( aa_nat_complex(F,N3) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F),groups2073611262835488442omplex(F,N6))) ) ) ).

% sums_finite
tff(fact_7912_sums__finite,axiom,
    ! [N6: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N6))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N6))
           => ( aa_nat_int(F,N3) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),groups3539618377306564664at_int(F,N6))) ) ) ).

% sums_finite
tff(fact_7913_sums__finite,axiom,
    ! [N6: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N6))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N6))
           => ( aa_nat_nat(F,N3) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N6))) ) ) ).

% sums_finite
tff(fact_7914_sums__finite,axiom,
    ! [N6: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N6))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N6))
           => ( aa_nat_real(F,N3) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),groups6591440286371151544t_real(F,N6))) ) ) ).

% sums_finite
tff(fact_7915_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_wb(P),F)),groups2073611262835488442omplex(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7916_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_wc(P),F)),groups3539618377306564664at_int(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7917_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wd(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7918_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_we(P),F)),groups6591440286371151544t_real(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7919_sums__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_wf(A),F)),groups2073611262835488442omplex(F,A))) ) ).

% sums_If_finite_set
tff(fact_7920_sums__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_wg(A),F)),groups3539618377306564664at_int(F,A))) ) ).

% sums_If_finite_set
tff(fact_7921_sums__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wh(A),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A))) ) ).

% sums_If_finite_set
tff(fact_7922_sums__If__finite__set,axiom,
    ! [A: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_wi(A),F)),groups6591440286371151544t_real(F,A))) ) ).

% sums_If_finite_set
tff(fact_7923_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_wj(M),Z)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),M))) ).

% powser_sums_if
tff(fact_7924_powser__sums__if,axiom,
    ! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_wk(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M))) ).

% powser_sums_if
tff(fact_7925_powser__sums__if,axiom,
    ! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_wl(M),Z)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),M))) ).

% powser_sums_if
tff(fact_7926_sums__iff__shift,axiom,
    ! [F: fun_nat_real,N: nat,S2: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ps(F),N)),S2))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_iff_shift
tff(fact_7927_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S2: real,N: nat] :
      ( pp(aa_real_bool(sums_real(F),S2))
     => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ps(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_split_initial_segment
tff(fact_7928_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,N: nat,S2: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ps(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N)))))
    <=> pp(aa_real_bool(sums_real(F),S2)) ) ).

% sums_iff_shift'
tff(fact_7929_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,Z),aa_nat_nat(suc,N))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups6464643781859351333omplex(aa_com8181001469867462503omplex(aTP_Lamm_wm,Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7930_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_wn(Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7931_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_wo,Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7932_pochhammer__code,axiom,
    ! [N: nat,A2: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A2),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A2),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_wp(A2),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_7933_pochhammer__code,axiom,
    ! [N: nat,A2: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_wq(A2),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_7934_pochhammer__code,axiom,
    ! [N: nat,A2: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),N) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_wr(A2),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_7935_pochhammer__code,axiom,
    ! [N: nat,A2: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ws(A2),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_7936_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_wt),N),zero_zero_complex) ).

% of_nat_code
tff(fact_7937_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_wu,N),zero_zero_rat) ).

% of_nat_code
tff(fact_7938_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_wv),N),zero_zero_real) ).

% of_nat_code
tff(fact_7939_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_ww),N),zero_zero_int) ).

% of_nat_code
tff(fact_7940_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_wx),N),zero_zero_nat) ).

% of_nat_code
tff(fact_7941_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_7942_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_wy,R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7943_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_wz(R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7944_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xa,R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7945_prod_Oneutral__const,axiom,
    ! [A: set_nat] : groups705719431365010083at_int(aTP_Lamm_xb,A) = one_one_int ).

% prod.neutral_const
tff(fact_7946_prod_Oneutral__const,axiom,
    ! [A: set_int] : groups1705073143266064639nt_int(aTP_Lamm_xc,A) = one_one_int ).

% prod.neutral_const
tff(fact_7947_prod_Oneutral__const,axiom,
    ! [A: set_nat] : groups708209901874060359at_nat(aTP_Lamm_xd,A) = one_one_nat ).

% prod.neutral_const
tff(fact_7948_of__nat__prod,axiom,
    ! [F: fun_int_nat,A: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A)) = groups1705073143266064639nt_int(aTP_Lamm_kb(F),A) ).

% of_nat_prod
tff(fact_7949_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_real(semiri5074537144036343181t_real,groups708209901874060359at_nat(F,A)) = groups129246275422532515t_real(aTP_Lamm_ke(F),A) ).

% of_nat_prod
tff(fact_7950_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_int(semiri1314217659103216013at_int,groups708209901874060359at_nat(F,A)) = groups705719431365010083at_int(aTP_Lamm_kc(F),A) ).

% of_nat_prod
tff(fact_7951_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,groups708209901874060359at_nat(F,A)) = groups708209901874060359at_nat(aTP_Lamm_kd(F),A) ).

% of_nat_prod
tff(fact_7952_of__int__prod,axiom,
    ! [F: fun_nat_int,A: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A)) = groups129246275422532515t_real(aTP_Lamm_kg(F),A) ).

% of_int_prod
tff(fact_7953_of__int__prod,axiom,
    ! [F: fun_nat_int,A: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A)) = groups73079841787564623at_rat(aTP_Lamm_xe(F),A) ).

% of_int_prod
tff(fact_7954_of__int__prod,axiom,
    ! [F: fun_nat_int,A: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A)) = groups705719431365010083at_int(aTP_Lamm_xf(F),A) ).

% of_int_prod
tff(fact_7955_of__int__prod,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A)) = groups2316167850115554303t_real(aTP_Lamm_kh(F),A) ).

% of_int_prod
tff(fact_7956_of__int__prod,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A)) = groups1072433553688619179nt_rat(aTP_Lamm_ki(F),A) ).

% of_int_prod
tff(fact_7957_of__int__prod,axiom,
    ! [F: fun_int_int,A: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A)) = groups1705073143266064639nt_int(aTP_Lamm_kj(F),A) ).

% of_int_prod
tff(fact_7958_prod_Oempty,axiom,
    ! [G: fun_nat_assn] : groups6906906614972039071t_assn(G,bot_bot_set_nat) = one_one_assn ).

% prod.empty
tff(fact_7959_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_7960_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_7961_prod_Oempty,axiom,
    ! [G: fun_int_assn] : groups7882442080178216443t_assn(G,bot_bot_set_int) = one_one_assn ).

% prod.empty
tff(fact_7962_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_7963_prod_Oempty,axiom,
    ! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).

% prod.empty
tff(fact_7964_prod_Oempty,axiom,
    ! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).

% prod.empty
tff(fact_7965_prod_Oempty,axiom,
    ! [G: fun_nat_int] : groups705719431365010083at_int(G,bot_bot_set_nat) = one_one_int ).

% prod.empty
tff(fact_7966_prod_Oempty,axiom,
    ! [G: fun_int_int] : groups1705073143266064639nt_int(G,bot_bot_set_int) = one_one_int ).

% prod.empty
tff(fact_7967_prod_Oempty,axiom,
    ! [G: fun_nat_nat] : groups708209901874060359at_nat(G,bot_bot_set_nat) = one_one_nat ).

% prod.empty
tff(fact_7968_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_assn] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups6906906614972039071t_assn(G,A) = one_one_assn ) ) ).

% prod.infinite
tff(fact_7969_prod_Oinfinite,axiom,
    ! [A: set_int,G: fun_int_assn] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups7882442080178216443t_assn(G,A) = one_one_assn ) ) ).

% prod.infinite
tff(fact_7970_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups1304777262505850412r_assn(G,A) = one_one_assn ) ) ).

% prod.infinite
tff(fact_7971_prod_Oinfinite,axiom,
    ! [A: set_complex,G: fun_complex_assn] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups4150731942483176573x_assn(G,A) = one_one_assn ) ) ).

% prod.infinite
tff(fact_7972_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups129246275422532515t_real(G,A) = one_one_real ) ) ).

% prod.infinite
tff(fact_7973_prod_Oinfinite,axiom,
    ! [A: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups2316167850115554303t_real(G,A) = one_one_real ) ) ).

% prod.infinite
tff(fact_7974_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups9004974159866482096r_real(G,A) = one_one_real ) ) ).

% prod.infinite
tff(fact_7975_prod_Oinfinite,axiom,
    ! [A: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups766887009212190081x_real(G,A) = one_one_real ) ) ).

% prod.infinite
tff(fact_7976_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups73079841787564623at_rat(G,A) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7977_prod_Oinfinite,axiom,
    ! [A: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups1072433553688619179nt_rat(G,A) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7978_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_7979_floor__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_7980_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_7981_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_7982_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_7983_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_7984_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_7985_gbinomial__0_I1_J,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_7986_gbinomial__0_I1_J,axiom,
    ! [A2: rat] : aa_nat_rat(gbinomial_rat(A2),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_7987_gbinomial__0_I1_J,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A2),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_7988_gbinomial__0_I1_J,axiom,
    ! [A2: int] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A2),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_7989_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_7990_floor__one,axiom,
    aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).

% floor_one
tff(fact_7991_prod_Odelta,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_xg(A2),B2),S) = aa_VEBT_VEBT_assn(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_xg(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta
tff(fact_7992_prod_Odelta,axiom,
    ! [S: set_real,A2: real,B2: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_xh(A2),B2),S) = aa_real_assn(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_xh(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta
tff(fact_7993_prod_Odelta,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_assn] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xi(A2),B2),S) = aa_nat_assn(B2,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xi(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta
tff(fact_7994_prod_Odelta,axiom,
    ! [S: set_int,A2: int,B2: fun_int_assn] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xj(A2),B2),S) = aa_int_assn(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xj(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta
tff(fact_7995_prod_Odelta,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xk(A2),B2),S) = aa_Code_integer_assn(B2,A2) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xk(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta
tff(fact_7996_prod_Odelta,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_xl(A2),B2),S) = aa_complex_assn(B2,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_xl(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta
tff(fact_7997_prod_Odelta,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_xm(A2),B2),S) = aa_VEBT_VEBT_real(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_xm(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7998_prod_Odelta,axiom,
    ! [S: set_real,A2: real,B2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_xn(A2),B2),S) = aa_real_real(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_xn(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7999_prod_Odelta,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_xo(A2),B2),S) = aa_nat_real(B2,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_xo(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8000_prod_Odelta,axiom,
    ! [S: set_int,A2: int,B2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_xp(A2),B2),S) = aa_int_real(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_xp(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_8001_prod_Odelta_H,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_xq(A2),B2),S) = aa_VEBT_VEBT_assn(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_xq(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta'
tff(fact_8002_prod_Odelta_H,axiom,
    ! [S: set_real,A2: real,B2: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_xr(A2),B2),S) = aa_real_assn(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_xr(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta'
tff(fact_8003_prod_Odelta_H,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_assn] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xs(A2),B2),S) = aa_nat_assn(B2,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xs(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta'
tff(fact_8004_prod_Odelta_H,axiom,
    ! [S: set_int,A2: int,B2: fun_int_assn] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xt(A2),B2),S) = aa_int_assn(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xt(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta'
tff(fact_8005_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xu(A2),B2),S) = aa_Code_integer_assn(B2,A2) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xu(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta'
tff(fact_8006_prod_Odelta_H,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_xv(A2),B2),S) = aa_complex_assn(B2,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_xv(A2),B2),S) = one_one_assn ) ) ) ) ).

% prod.delta'
tff(fact_8007_prod_Odelta_H,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_xw(A2),B2),S) = aa_VEBT_VEBT_real(B2,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_xw(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8008_prod_Odelta_H,axiom,
    ! [S: set_real,A2: real,B2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_xx(A2),B2),S) = aa_real_real(B2,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_xx(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8009_prod_Odelta_H,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_xy(A2),B2),S) = aa_nat_real(B2,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_xy(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8010_prod_Odelta_H,axiom,
    ! [S: set_int,A2: int,B2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_xz(A2),B2),S) = aa_int_real(B2,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_xz(A2),B2),S) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_8011_prod_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,A)) ) ) ) ).

% prod.insert
tff(fact_8012_prod_Oinsert,axiom,
    ! [A: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A))
       => ( groups1681761925125756287l_real(G,insert_real(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,A)) ) ) ) ).

% prod.insert
tff(fact_8013_prod_Oinsert,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
       => ( groups129246275422532515t_real(G,insert_nat(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),groups129246275422532515t_real(G,A)) ) ) ) ).

% prod.insert
tff(fact_8014_prod_Oinsert,axiom,
    ! [A: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A))
       => ( groups2316167850115554303t_real(G,insert_int(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,A)) ) ) ) ).

% prod.insert
tff(fact_8015_prod_Oinsert,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups9004974159866482096r_real(G,insert_Code_integer(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,X2)),groups9004974159866482096r_real(G,A)) ) ) ) ).

% prod.insert
tff(fact_8016_prod_Oinsert,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ~ pp(aa_set_complex_bool(member_complex(X2),A))
       => ( groups766887009212190081x_real(G,insert_complex(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,A)) ) ) ) ).

% prod.insert
tff(fact_8017_prod_Oinsert,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_8018_prod_Oinsert,axiom,
    ! [A: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A))
       => ( groups4061424788464935467al_rat(G,insert_real(X2,A)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_8019_prod_Oinsert,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
       => ( groups73079841787564623at_rat(G,insert_nat(X2,A)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_8020_prod_Oinsert,axiom,
    ! [A: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A))
       => ( groups1072433553688619179nt_rat(G,insert_int(X2,A)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_8021_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% prod.lessThan_Suc
tff(fact_8022_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).

% prod.lessThan_Suc
tff(fact_8023_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% prod.lessThan_Suc
tff(fact_8024_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% prod.lessThan_Suc
tff(fact_8025_floor__diff__of__int,axiom,
    ! [X2: real,Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X2)),Z) ).

% floor_diff_of_int
tff(fact_8026_floor__diff__of__int,axiom,
    ! [X2: rat,Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X2),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),Z) ).

% floor_diff_of_int
tff(fact_8027_zero__le__floor,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% zero_le_floor
tff(fact_8028_zero__le__floor,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2)) ) ).

% zero_le_floor
tff(fact_8029_numeral__le__floor,axiom,
    ! [V: num,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X2)) ) ).

% numeral_le_floor
tff(fact_8030_numeral__le__floor,axiom,
    ! [V: num,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X2)) ) ).

% numeral_le_floor
tff(fact_8031_floor__less__zero,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X2)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% floor_less_zero
tff(fact_8032_floor__less__zero,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X2)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat)) ) ).

% floor_less_zero
tff(fact_8033_floor__less__numeral,axiom,
    ! [X2: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_numeral
tff(fact_8034_floor__less__numeral,axiom,
    ! [X2: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_numeral
tff(fact_8035_zero__less__floor,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ).

% zero_less_floor
tff(fact_8036_zero__less__floor,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X2)) ) ).

% zero_less_floor
tff(fact_8037_floor__le__zero,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ).

% floor_le_zero
tff(fact_8038_floor__le__zero,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),one_one_rat)) ) ).

% floor_le_zero
tff(fact_8039_one__le__floor,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ).

% one_le_floor
tff(fact_8040_one__le__floor,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X2)) ) ).

% one_le_floor
tff(fact_8041_floor__less__one,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X2)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ).

% floor_less_one
tff(fact_8042_floor__less__one,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X2)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),one_one_rat)) ) ).

% floor_less_one
tff(fact_8043_floor__diff__numeral,axiom,
    ! [X2: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_8044_floor__diff__numeral,axiom,
    ! [X2: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X2),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_8045_floor__numeral__power,axiom,
    ! [X2: num,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X2)),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ).

% floor_numeral_power
tff(fact_8046_floor__numeral__power,axiom,
    ! [X2: num,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X2)),N) ).

% floor_numeral_power
tff(fact_8047_floor__diff__one,axiom,
    ! [X2: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X2)),one_one_int) ).

% floor_diff_one
tff(fact_8048_floor__diff__one,axiom,
    ! [X2: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X2),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),one_one_int) ).

% floor_diff_one
tff(fact_8049_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_assn] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6906906614972039071t_assn(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = one_one_assn ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6906906614972039071t_assn(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,set_or4665077453230672383an_nat(M,N))),aa_nat_assn(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8050_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,N))),aa_nat_real(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8051_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8052_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,N))),aa_nat_int(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8053_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_nat(G,N)) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_8054_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_assn] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6906906614972039071t_assn(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = one_one_assn ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6906906614972039071t_assn(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,set_or1269000886237332187st_nat(M,N))),aa_nat_assn(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8055_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8056_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8057_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8058_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8059_numeral__less__floor,axiom,
    ! [V: num,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X2)) ) ).

% numeral_less_floor
tff(fact_8060_numeral__less__floor,axiom,
    ! [V: num,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X2)) ) ).

% numeral_less_floor
tff(fact_8061_floor__le__numeral,axiom,
    ! [X2: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% floor_le_numeral
tff(fact_8062_floor__le__numeral,axiom,
    ! [X2: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% floor_le_numeral
tff(fact_8063_one__less__floor,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)) ) ).

% one_less_floor
tff(fact_8064_one__less__floor,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X2)) ) ).

% one_less_floor
tff(fact_8065_floor__le__one,axiom,
    ! [X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_8066_floor__le__one,axiom,
    ! [X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_8067_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8068_prod_Oneutral,axiom,
    ! [A: set_nat,G: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => ( aa_nat_int(G,X4) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A) = one_one_int ) ) ).

% prod.neutral
tff(fact_8069_prod_Oneutral,axiom,
    ! [A: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => ( aa_int_int(G,X4) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A) = one_one_int ) ) ).

% prod.neutral
tff(fact_8070_prod_Oneutral,axiom,
    ! [A: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => ( aa_nat_nat(G,X4) = one_one_nat ) )
     => ( groups708209901874060359at_nat(G,A) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8071_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_assn,A: set_nat] :
      ( ( groups6906906614972039071t_assn(G,A) != one_one_assn )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A))
           => ( aa_nat_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8072_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_assn,A: set_VEBT_VEBT] :
      ( ( groups569574905686396791T_assn(G,A) != one_one_assn )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
           => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8073_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_assn,A: set_real] :
      ( ( groups1155561341820557179l_assn(G,A) != one_one_assn )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A))
           => ( aa_real_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8074_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_assn,A: set_int] :
      ( ( groups7882442080178216443t_assn(G,A) != one_one_assn )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A))
           => ( aa_int_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8075_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A: set_nat] :
      ( ( groups129246275422532515t_real(G,A) != one_one_real )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A))
           => ( aa_nat_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8076_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A: set_VEBT_VEBT] :
      ( ( groups2703838992350267259T_real(G,A) != one_one_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
           => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8077_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A: set_real] :
      ( ( groups1681761925125756287l_real(G,A) != one_one_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A))
           => ( aa_real_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8078_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A: set_int] :
      ( ( groups2316167850115554303t_real(G,A) != one_one_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A))
           => ( aa_int_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8079_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A: set_nat] :
      ( ( groups73079841787564623at_rat(G,A) != one_one_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A))
           => ( aa_nat_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8080_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_rat,A: set_VEBT_VEBT] :
      ( ( groups5726676334696518183BT_rat(G,A) != one_one_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
           => ( aa_VEBT_VEBT_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8081_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_int,B: set_nat,A: set_nat] : groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_ya(G),B),A) = groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_yc(G),A),B) ).

% prod.swap
tff(fact_8082_prod_Oswap,axiom,
    ! [G: fun_nat_fun_int_int,B: set_int,A: set_nat] : groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_yd(G),B),A) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_yf(G),A),B) ).

% prod.swap
tff(fact_8083_prod_Oswap,axiom,
    ! [G: fun_int_fun_nat_int,B: set_nat,A: set_int] : groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_yg(G),B),A) = groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_yh(G),A),B) ).

% prod.swap
tff(fact_8084_prod_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B: set_int,A: set_int] : groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_yi(G),B),A) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_yj(G),A),B) ).

% prod.swap
tff(fact_8085_prod_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B: set_nat,A: set_nat] : groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_yk(G),B),A) = groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_yl(G),A),B) ).

% prod.swap
tff(fact_8086_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_rn(G),H),A) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A)),groups705719431365010083at_int(H,A)) ).

% prod.distrib
tff(fact_8087_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ym(G),H),A) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A)),groups1705073143266064639nt_int(H,A)) ).

% prod.distrib
tff(fact_8088_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rp(G),H),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A)),groups708209901874060359at_nat(H,A)) ).

% prod.distrib
tff(fact_8089_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A: set_nat,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups705719431365010083at_int(F,A)),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_yn(F),N),A) ).

% prod_power_distrib
tff(fact_8090_prod__power__distrib,axiom,
    ! [F: fun_int_int,A: set_int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups1705073143266064639nt_int(F,A)),N) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_yo(F),N),A) ).

% prod_power_distrib
tff(fact_8091_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A: set_nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A)),N) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_yp(F),N),A) ).

% prod_power_distrib
tff(fact_8092_prod_Oswap__restrict,axiom,
    ! [A: set_VEBT_VEBT,B: set_nat,G: fun_VE7331017413731640256at_int,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups6359315924273963643BT_int(aa_fun6141810409722670662BT_int(aa_fun6474440919679335769BT_int(aTP_Lamm_yq(B),G),R),A) = groups705719431365010083at_int(aa_fun1384843155787029862at_int(aa_fun6066863168309489065at_int(aTP_Lamm_ys(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8093_prod_Oswap__restrict,axiom,
    ! [A: set_real,B: set_nat,G: fun_real_fun_nat_int,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups4694064378042380927al_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_yt(B),G),R),A) = groups705719431365010083at_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_yv(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8094_prod_Oswap__restrict,axiom,
    ! [A: set_Code_integer,B: set_nat,G: fun_Co822469698833053651at_int,R: fun_Co8657364630979331932t_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups3188404863801439024er_int(aa_fun7994889559813147430er_int(aa_fun6206124202175859492er_int(aTP_Lamm_yw(B),G),R),A) = groups705719431365010083at_int(aa_fun3371163242325571225at_int(aa_fun5074241136823302423at_int(aTP_Lamm_yy(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8095_prod_Oswap__restrict,axiom,
    ! [A: set_complex,B: set_nat,G: fun_co7370231399432144932at_int,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups858564598930262913ex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_yz(B),G),R),A) = groups705719431365010083at_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_zb(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8096_prod_Oswap__restrict,axiom,
    ! [A: set_VEBT_VEBT,B: set_int,G: fun_VE294566951569886236nt_int,R: fun_VE3303673267288458579t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups6359315924273963643BT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_zd(B),G),R),A) = groups1705073143266064639nt_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_zf(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8097_prod_Oswap__restrict,axiom,
    ! [A: set_real,B: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups4694064378042380927al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_zh(B),G),R),A) = groups1705073143266064639nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_zj(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8098_prod_Oswap__restrict,axiom,
    ! [A: set_Code_integer,B: set_int,G: fun_Co3009391273526075439nt_int,R: fun_Co8197081679332685312t_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups3188404863801439024er_int(aa_fun531554186711482826er_int(aa_fun2196751070425669740er_int(aTP_Lamm_zl(B),G),R),A) = groups1705073143266064639nt_int(aa_fun7408359038690759577nt_int(aa_fun3657818423959754427nt_int(aTP_Lamm_zo(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8099_prod_Oswap__restrict,axiom,
    ! [A: set_complex,B: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_int_bool(finite_finite_int,B))
       => ( groups858564598930262913ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_zq(B),G),R),A) = groups1705073143266064639nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_zt(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8100_prod_Oswap__restrict,axiom,
    ! [A: set_VEBT_VEBT,B: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups6361806394783013919BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_zu(B),G),R),A) = groups708209901874060359at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_zv(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8101_prod_Oswap__restrict,axiom,
    ! [A: set_real,B: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B))
       => ( groups4696554848551431203al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_zw(B),G),R),A) = groups708209901874060359at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_zx(A),G),R),B) ) ) ) ).

% prod.swap_restrict
tff(fact_8102_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A2: int,A: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_zy(F),A2),A)),A2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(F,A)),A2) ).

% mod_prod_eq
tff(fact_8103_mod__prod__eq,axiom,
    ! [F: fun_int_int,A2: int,A: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ot(F),A2),A)),A2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A)),A2) ).

% mod_prod_eq
tff(fact_8104_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A2: nat,A: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_os(F),A2),A)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(F,A)),A2) ).

% mod_prod_eq
tff(fact_8105_prod__nonneg,axiom,
    ! [A: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A))) ) ).

% prod_nonneg
tff(fact_8106_prod__nonneg,axiom,
    ! [A: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A))) ) ).

% prod_nonneg
tff(fact_8107_prod__nonneg,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups708209901874060359at_nat(F,A))) ) ).

% prod_nonneg
tff(fact_8108_prod__mono,axiom,
    ! [A: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),aa_nat_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A)),groups129246275422532515t_real(G,A))) ) ).

% prod_mono
tff(fact_8109_prod__mono,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I2)),aa_VEBT_VEBT_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A)),groups2703838992350267259T_real(G,A))) ) ).

% prod_mono
tff(fact_8110_prod__mono,axiom,
    ! [A: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A)),groups1681761925125756287l_real(G,A))) ) ).

% prod_mono
tff(fact_8111_prod__mono,axiom,
    ! [A: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A)),groups2316167850115554303t_real(G,A))) ) ).

% prod_mono
tff(fact_8112_prod__mono,axiom,
    ! [A: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A)),groups73079841787564623at_rat(G,A))) ) ).

% prod_mono
tff(fact_8113_prod__mono,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A)),groups5726676334696518183BT_rat(G,A))) ) ).

% prod_mono
tff(fact_8114_prod__mono,axiom,
    ! [A: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A)),groups4061424788464935467al_rat(G,A))) ) ).

% prod_mono
tff(fact_8115_prod__mono,axiom,
    ! [A: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A)),groups1072433553688619179nt_rat(G,A))) ) ).

% prod_mono
tff(fact_8116_prod__mono,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups6361806394783013919BT_nat(F,A)),groups6361806394783013919BT_nat(G,A))) ) ).

% prod_mono
tff(fact_8117_prod__mono,axiom,
    ! [A: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A)),groups4696554848551431203al_nat(G,A))) ) ).

% prod_mono
tff(fact_8118_prod__pos,axiom,
    ! [A: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A))) ) ).

% prod_pos
tff(fact_8119_prod__pos,axiom,
    ! [A: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A))) ) ).

% prod_pos
tff(fact_8120_prod__pos,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A))) ) ).

% prod_pos
tff(fact_8121_prod__ge__1,axiom,
    ! [A: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A))) ) ).

% prod_ge_1
tff(fact_8122_prod__ge__1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2703838992350267259T_real(F,A))) ) ).

% prod_ge_1
tff(fact_8123_prod__ge__1,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A))) ) ).

% prod_ge_1
tff(fact_8124_prod__ge__1,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A))) ) ).

% prod_ge_1
tff(fact_8125_prod__ge__1,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A))) ) ).

% prod_ge_1
tff(fact_8126_prod__ge__1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups5726676334696518183BT_rat(F,A))) ) ).

% prod_ge_1
tff(fact_8127_prod__ge__1,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A))) ) ).

% prod_ge_1
tff(fact_8128_prod__ge__1,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A))) ) ).

% prod_ge_1
tff(fact_8129_prod__ge__1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups6361806394783013919BT_nat(F,A))) ) ).

% prod_ge_1
tff(fact_8130_prod__ge__1,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A))) ) ).

% prod_ge_1
tff(fact_8131_of__int__floor__le,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X2))),X2)) ).

% of_int_floor_le
tff(fact_8132_of__int__floor__le,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X2))),X2)) ).

% of_int_floor_le
tff(fact_8133_floor__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim6058952711729229775r_real,Y3))) ) ).

% floor_mono
tff(fact_8134_floor__mono,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim3151403230148437115or_rat,Y3))) ) ).

% floor_mono
tff(fact_8135_floor__less__cancel,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim6058952711729229775r_real,Y3)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ).

% floor_less_cancel
tff(fact_8136_floor__less__cancel,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim3151403230148437115or_rat,Y3)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3)) ) ).

% floor_less_cancel
tff(fact_8137_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_assn,A2: nat,B2: nat] : groups6906906614972039071t_assn(F,set_or1269000886237332187st_nat(A2,B2)) = set_fo1959793692361082170t_assn(aTP_Lamm_zz(F),A2,B2,one_one_assn) ).

% prod_atLeastAtMost_code
tff(fact_8138_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A2: nat,B2: nat] : groups129246275422532515t_real(F,set_or1269000886237332187st_nat(A2,B2)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_aaa(F),A2),B2),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8139_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A2: nat,B2: nat] : groups73079841787564623at_rat(F,set_or1269000886237332187st_nat(A2,B2)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_aab(F),A2,B2),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8140_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A2: nat,B2: nat] : groups705719431365010083at_int(F,set_or1269000886237332187st_nat(A2,B2)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_aac(F),A2),B2),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8141_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A2: nat,B2: nat] : groups708209901874060359at_nat(F,set_or1269000886237332187st_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_aad(F),A2),B2),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8142_floor__le__ceiling,axiom,
    ! [X2: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim7802044766580827645g_real,X2))) ).

% floor_le_ceiling
tff(fact_8143_floor__le__ceiling,axiom,
    ! [X2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim2889992004027027881ng_rat,X2))) ).

% floor_le_ceiling
tff(fact_8144_floor__le__round,axiom,
    ! [X2: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim8280529875227126926d_real,X2))) ).

% floor_le_round
tff(fact_8145_floor__le__round,axiom,
    ! [X2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim7778729529865785530nd_rat,X2))) ).

% floor_le_round
tff(fact_8146_prod_Ointer__filter,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups569574905686396791T_assn(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_au(A),P))) = groups569574905686396791T_assn(aa_fun158440353102879243T_assn(aTP_Lamm_aae(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8147_prod_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_assn,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( groups1155561341820557179l_assn(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_av(A),P))) = groups1155561341820557179l_assn(aa_fun179583594588223833l_assn(aTP_Lamm_aaf(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8148_prod_Ointer__filter,axiom,
    ! [A: set_nat,G: fun_nat_assn,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups6906906614972039071t_assn(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(A),P))) = groups6906906614972039071t_assn(aa_fun1954384836919635929t_assn(aTP_Lamm_aag(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8149_prod_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_assn,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups7882442080178216443t_assn(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_az(A),P))) = groups7882442080178216443t_assn(aa_fun6196142345855524569t_assn(aTP_Lamm_aah(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8150_prod_Ointer__filter,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups1304777262505850412r_assn(G,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_oy(A),P))) = groups1304777262505850412r_assn(aa_fun8033543692415806745r_assn(aTP_Lamm_aai(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8151_prod_Ointer__filter,axiom,
    ! [A: set_complex,G: fun_complex_assn,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups4150731942483176573x_assn(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ax(A),P))) = groups4150731942483176573x_assn(aa_fun3623554829249773913x_assn(aTP_Lamm_aaj(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8152_prod_Ointer__filter,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups2703838992350267259T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_au(A),P))) = groups2703838992350267259T_real(aa_fun2072160908118572431T_real(aTP_Lamm_aak(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8153_prod_Ointer__filter,axiom,
    ! [A: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_av(A),P))) = groups1681761925125756287l_real(aa_fun6348708079608492509l_real(aTP_Lamm_aal(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8154_prod_Ointer__filter,axiom,
    ! [A: set_nat,G: fun_nat_real,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups129246275422532515t_real(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(A),P))) = groups129246275422532515t_real(aa_fun1750223370215434333t_real(aTP_Lamm_aam(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8155_prod_Ointer__filter,axiom,
    ! [A: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_az(A),P))) = groups2316167850115554303t_real(aa_fun6974987029372589917t_real(aTP_Lamm_aan(G),P),A) ) ) ).

% prod.inter_filter
tff(fact_8156_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups705719431365010083at_int(aTP_Lamm_ql(G),set_or4665077453230672383an_nat(M,N)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_8157_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups708209901874060359at_nat(aTP_Lamm_pp(G),set_or4665077453230672383an_nat(M,N)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_8158_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups705719431365010083at_int(aTP_Lamm_ql(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8159_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups708209901874060359at_nat(aTP_Lamm_pp(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8160_power__sum,axiom,
    ! [C2: real,F: fun_nat_nat,A: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_aao(C2),F),A) ).

% power_sum
tff(fact_8161_power__sum,axiom,
    ! [C2: complex,F: fun_nat_nat,A: set_nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,C2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_aap(C2),F),A) ).

% power_sum
tff(fact_8162_power__sum,axiom,
    ! [C2: code_integer,F: fun_nat_nat,A: set_nat] : aa_nat_Code_integer(power_8256067586552552935nteger(C2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups3455450783089532116nteger(aa_fun8101275520843608965nteger(aTP_Lamm_aaq(C2),F),A) ).

% power_sum
tff(fact_8163_power__sum,axiom,
    ! [C2: int,F: fun_nat_nat,A: set_nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_aar(C2),F),A) ).

% power_sum
tff(fact_8164_power__sum,axiom,
    ! [C2: int,F: fun_int_nat,A: set_int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C2),groups4541462559716669496nt_nat(F,A)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_aas(C2),F),A) ).

% power_sum
tff(fact_8165_power__sum,axiom,
    ! [C2: nat,F: fun_nat_nat,A: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aat(C2),F),A) ).

% power_sum
tff(fact_8166_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : groups705719431365010083at_int(G,set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_aau(G),K),set_or4665077453230672383an_nat(M,N)) ).

% prod.shift_bounds_nat_ivl
tff(fact_8167_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr(G),K),set_or4665077453230672383an_nat(M,N)) ).

% prod.shift_bounds_nat_ivl
tff(fact_8168_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_aau(G),K),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8169_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr(G),K),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8170_prod__le__1,axiom,
    ! [A: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A)),one_one_real)) ) ).

% prod_le_1
tff(fact_8171_prod__le__1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A)),one_one_real)) ) ).

% prod_le_1
tff(fact_8172_prod__le__1,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A)),one_one_real)) ) ).

% prod_le_1
tff(fact_8173_prod__le__1,axiom,
    ! [A: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A)),one_one_real)) ) ).

% prod_le_1
tff(fact_8174_prod__le__1,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8175_prod__le__1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8176_prod__le__1,axiom,
    ! [A: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8177_prod__le__1,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8178_prod__le__1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups6361806394783013919BT_nat(F,A)),one_one_nat)) ) ).

% prod_le_1
tff(fact_8179_prod__le__1,axiom,
    ! [A: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A)),one_one_nat)) ) ).

% prod_le_1
tff(fact_8180_prod_Orelated,axiom,
    ! [R: fun_as7903617535776176121n_bool,S: set_nat,H: fun_nat_assn,G: fun_nat_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,one_one_assn),one_one_assn))
     => ( ! [X15: assn,Y1: assn,X23: assn,Y22: assn] :
            ( ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,X15),X23))
              & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,Y1),Y22)) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_assn_assn(times_times_assn(X15),Y1)),aa_assn_assn(times_times_assn(X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_nat_assn(H,X4)),aa_nat_assn(G,X4))) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,groups6906906614972039071t_assn(H,S)),groups6906906614972039071t_assn(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8181_prod_Orelated,axiom,
    ! [R: fun_as7903617535776176121n_bool,S: set_int,H: fun_int_assn,G: fun_int_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,one_one_assn),one_one_assn))
     => ( ! [X15: assn,Y1: assn,X23: assn,Y22: assn] :
            ( ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,X15),X23))
              & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,Y1),Y22)) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_assn_assn(times_times_assn(X15),Y1)),aa_assn_assn(times_times_assn(X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_int_assn(H,X4)),aa_int_assn(G,X4))) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,groups7882442080178216443t_assn(H,S)),groups7882442080178216443t_assn(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8182_prod_Orelated,axiom,
    ! [R: fun_as7903617535776176121n_bool,S: set_Code_integer,H: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,one_one_assn),one_one_assn))
     => ( ! [X15: assn,Y1: assn,X23: assn,Y22: assn] :
            ( ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,X15),X23))
              & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,Y1),Y22)) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_assn_assn(times_times_assn(X15),Y1)),aa_assn_assn(times_times_assn(X23),Y22))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_Code_integer_assn(H,X4)),aa_Code_integer_assn(G,X4))) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,groups1304777262505850412r_assn(H,S)),groups1304777262505850412r_assn(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8183_prod_Orelated,axiom,
    ! [R: fun_as7903617535776176121n_bool,S: set_complex,H: fun_complex_assn,G: fun_complex_assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,one_one_assn),one_one_assn))
     => ( ! [X15: assn,Y1: assn,X23: assn,Y22: assn] :
            ( ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,X15),X23))
              & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,Y1),Y22)) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_assn_assn(times_times_assn(X15),Y1)),aa_assn_assn(times_times_assn(X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,aa_complex_assn(H,X4)),aa_complex_assn(G,X4))) )
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(R,groups4150731942483176573x_assn(H,S)),groups4150731942483176573x_assn(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8184_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_nat_real(H,X4)),aa_nat_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups129246275422532515t_real(H,S)),groups129246275422532515t_real(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8185_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups2316167850115554303t_real(H,S)),groups2316167850115554303t_real(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8186_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y22))) )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Code_integer_real(H,X4)),aa_Code_integer_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups9004974159866482096r_real(H,S)),groups9004974159866482096r_real(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8187_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y1: real,X23: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups766887009212190081x_real(H,S)),groups766887009212190081x_real(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8188_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X15),Y1)),aa_rat_rat(times_times_rat(X23),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups73079841787564623at_rat(H,S)),groups73079841787564623at_rat(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8189_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X15),Y1)),aa_rat_rat(times_times_rat(X23),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups1072433553688619179nt_rat(H,S)),groups1072433553688619179nt_rat(G,S))) ) ) ) ) ).

% prod.related
tff(fact_8190_prod_Oivl__cong,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( ( A2 = C2 )
     => ( ( B2 = D2 )
       => ( ! [X4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X4))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D2))
               => ( aa_nat_int(G,X4) = aa_nat_int(H,X4) ) ) )
         => ( groups705719431365010083at_int(G,set_or4665077453230672383an_nat(A2,B2)) = groups705719431365010083at_int(H,set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8191_prod_Oivl__cong,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A2 = C2 )
     => ( ( B2 = D2 )
       => ( ! [X4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X4))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D2))
               => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) ) )
         => ( groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(A2,B2)) = groups708209901874060359at_nat(H,set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8192_prod_Oivl__cong,axiom,
    ! [A2: int,C2: int,B2: int,D2: int,G: fun_int_int,H: fun_int_int] :
      ( ( A2 = C2 )
     => ( ( B2 = D2 )
       => ( ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),X4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D2))
               => ( aa_int_int(G,X4) = aa_int_int(H,X4) ) ) )
         => ( groups1705073143266064639nt_int(G,set_or4662586982721622107an_int(A2,B2)) = groups1705073143266064639nt_int(H,set_or4662586982721622107an_int(C2,D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8193_prod_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A)) = groups2703838992350267259T_real(G,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8194_prod_Oinsert__if,axiom,
    ! [A: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( pp(aa_set_real_bool(member_real(X2),A))
         => ( groups1681761925125756287l_real(G,insert_real(X2,A)) = groups1681761925125756287l_real(G,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A))
         => ( groups1681761925125756287l_real(G,insert_real(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8195_prod_Oinsert__if,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( pp(aa_set_nat_bool(member_nat(X2),A))
         => ( groups129246275422532515t_real(G,insert_nat(X2,A)) = groups129246275422532515t_real(G,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
         => ( groups129246275422532515t_real(G,insert_nat(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),groups129246275422532515t_real(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8196_prod_Oinsert__if,axiom,
    ! [A: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( pp(aa_set_int_bool(member_int(X2),A))
         => ( groups2316167850115554303t_real(G,insert_int(X2,A)) = groups2316167850115554303t_real(G,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A))
         => ( groups2316167850115554303t_real(G,insert_int(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8197_prod_Oinsert__if,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
         => ( groups9004974159866482096r_real(G,insert_Code_integer(X2,A)) = groups9004974159866482096r_real(G,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
         => ( groups9004974159866482096r_real(G,insert_Code_integer(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,X2)),groups9004974159866482096r_real(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8198_prod_Oinsert__if,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( pp(aa_set_complex_bool(member_complex(X2),A))
         => ( groups766887009212190081x_real(G,insert_complex(X2,A)) = groups766887009212190081x_real(G,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X2),A))
         => ( groups766887009212190081x_real(G,insert_complex(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8199_prod_Oinsert__if,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A)) = groups5726676334696518183BT_rat(G,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
         => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8200_prod_Oinsert__if,axiom,
    ! [A: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( pp(aa_set_real_bool(member_real(X2),A))
         => ( groups4061424788464935467al_rat(G,insert_real(X2,A)) = groups4061424788464935467al_rat(G,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A))
         => ( groups4061424788464935467al_rat(G,insert_real(X2,A)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8201_prod_Oinsert__if,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( pp(aa_set_nat_bool(member_nat(X2),A))
         => ( groups73079841787564623at_rat(G,insert_nat(X2,A)) = groups73079841787564623at_rat(G,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A))
         => ( groups73079841787564623at_rat(G,insert_nat(X2,A)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8202_prod_Oinsert__if,axiom,
    ! [A: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( pp(aa_set_int_bool(member_int(X2),A))
         => ( groups1072433553688619179nt_rat(G,insert_int(X2,A)) = groups1072433553688619179nt_rat(G,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A))
         => ( groups1072433553688619179nt_rat(G,insert_int(X2,A)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,A)) ) ) ) ) ).

% prod.insert_if
tff(fact_8203_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_VEBT_VEBT,S: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T5: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T5,T6))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),T6))
                       => ( aa_VEBT_VEBT_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_VEBT_VEBT_assn(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_assn(G,A3) ) )
                     => ( groups569574905686396791T_assn(G,S) = groups569574905686396791T_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8204_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_real,S: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T5: set_real,G: fun_VEBT_VEBT_assn,H: fun_real_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set_real_bool(finite_finite_real,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T5,T6))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T6))
                       => ( aa_real_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_real_assn(H,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_assn(G,A3) ) )
                     => ( groups569574905686396791T_assn(G,S) = groups1155561341820557179l_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8205_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_VEBT_VEBT,S: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T5: set_VEBT_VEBT,G: fun_real_assn,H: fun_VEBT_VEBT_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T5,T6))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),T6))
                       => ( aa_VEBT_VEBT_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_VEBT_VEBT_assn(H,aa_real_VEBT_VEBT(J,A3)) = aa_real_assn(G,A3) ) )
                     => ( groups1155561341820557179l_assn(G,S) = groups569574905686396791T_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8206_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_real,S: set_real,I: fun_real_real,J: fun_real_real,T5: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set_real_bool(finite_finite_real,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T5,T6))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T6))
                       => ( aa_real_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_real_assn(H,aa_real_real(J,A3)) = aa_real_assn(G,A3) ) )
                     => ( groups1155561341820557179l_assn(G,S) = groups1155561341820557179l_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8207_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_int,S: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T5: set_int,G: fun_VEBT_VEBT_assn,H: fun_int_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set_int_bool(finite_finite_int,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T5,T6))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T6))
                       => ( aa_int_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_int_assn(H,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_assn(G,A3) ) )
                     => ( groups569574905686396791T_assn(G,S) = groups7882442080178216443t_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8208_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_int,S: set_real,I: fun_int_real,J: fun_real_int,T5: set_int,G: fun_real_assn,H: fun_int_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set_int_bool(finite_finite_int,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T5,T6))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T6))
                       => ( aa_int_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_int_assn(H,aa_real_int(J,A3)) = aa_real_assn(G,A3) ) )
                     => ( groups1155561341820557179l_assn(G,S) = groups7882442080178216443t_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8209_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_Code_integer,S: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T5: set_Code_integer,G: fun_VEBT_VEBT_assn,H: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set1374477347934578310r_bool(member_Code_integer(aa_VEB5935388676523804373nteger(J,A3)),minus_2355218937544613996nteger(T5,T6))) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),T6))
                       => ( aa_Code_integer_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_Code_integer_assn(H,aa_VEB5935388676523804373nteger(J,A3)) = aa_VEBT_VEBT_assn(G,A3) ) )
                     => ( groups569574905686396791T_assn(G,S) = groups1304777262505850412r_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8210_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_Code_integer,S: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T5: set_Code_integer,G: fun_real_assn,H: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set1374477347934578310r_bool(member_Code_integer(aa_real_Code_integer(J,A3)),minus_2355218937544613996nteger(T5,T6))) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B3)) = B3 ) )
             => ( ! [B3: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_Code_integer_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),T6))
                       => ( aa_Code_integer_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_Code_integer_assn(H,aa_real_Code_integer(J,A3)) = aa_real_assn(G,A3) ) )
                     => ( groups1155561341820557179l_assn(G,S) = groups1304777262505850412r_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8211_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T6: set_complex,S: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T5: set_complex,G: fun_VEBT_VEBT_assn,H: fun_complex_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S6))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T6))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S,S6)))
               => pp(aa_set_complex_bool(member_complex(aa_VEBT_VEBT_complex(J,A3)),minus_811609699411566653omplex(T5,T6))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T5,T6)))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T5,T6)))
                   => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_complex_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S,S6))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S6))
                     => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T6))
                       => ( aa_complex_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),S))
                         => ( aa_complex_assn(H,aa_VEBT_VEBT_complex(J,A3)) = aa_VEBT_VEBT_assn(G,A3) ) )
                     => ( groups569574905686396791T_assn(G,S) = groups4150731942483176573x_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8212_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T6: set_complex,S: set_real,I: fun_complex_real,J: fun_real_complex,T5: set_complex,G: fun_real_assn,H: fun_complex_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,S6))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T6))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S,S6)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T5,T6))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T5,T6)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T5,T6)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S,S6))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S6))
                     => ( aa_real_assn(G,A3) = one_one_assn ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T6))
                       => ( aa_complex_assn(H,B3) = one_one_assn ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S))
                         => ( aa_complex_assn(H,aa_real_complex(J,A3)) = aa_real_assn(G,A3) ) )
                     => ( groups1155561341820557179l_assn(G,S) = groups4150731942483176573x_assn(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8213_prod__dvd__prod__subset2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A3)),aa_VEBT_VEBT_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups6361806394783013919BT_nat(F,A)),groups6361806394783013919BT_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8214_prod__dvd__prod__subset2,axiom,
    ! [B: set_real,A: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups4696554848551431203al_nat(F,A)),groups4696554848551431203al_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8215_prod__dvd__prod__subset2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( ! [A3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_Code_integer_nat(F,A3)),aa_Code_integer_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups3190895334310489300er_nat(F,A)),groups3190895334310489300er_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8216_prod__dvd__prod__subset2,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A)),groups861055069439313189ex_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8217_prod__dvd__prod__subset2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_VEBT_VEBT_int(F,A3)),aa_VEBT_VEBT_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups6359315924273963643BT_int(F,A)),groups6359315924273963643BT_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8218_prod__dvd__prod__subset2,axiom,
    ! [B: set_real,A: set_real,F: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups4694064378042380927al_int(F,A)),groups4694064378042380927al_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8219_prod__dvd__prod__subset2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( ! [A3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_Code_integer_int(F,A3)),aa_Code_integer_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups3188404863801439024er_int(F,A)),groups3188404863801439024er_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8220_prod__dvd__prod__subset2,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A)),groups858564598930262913ex_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8221_prod__dvd__prod__subset2,axiom,
    ! [B: set_int,A: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A3)),aa_int_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A)),groups1707563613775114915nt_nat(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8222_prod__dvd__prod__subset2,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A3)),aa_nat_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups705719431365010083at_int(F,A)),groups705719431365010083at_int(G,B))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8223_prod__dvd__prod__subset,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups3190895334310489300er_nat(F,A)),groups3190895334310489300er_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8224_prod__dvd__prod__subset,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A)),groups861055069439313189ex_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8225_prod__dvd__prod__subset,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups3188404863801439024er_int(F,A)),groups3188404863801439024er_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8226_prod__dvd__prod__subset,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A)),groups858564598930262913ex_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8227_prod__dvd__prod__subset,axiom,
    ! [B: set_int,A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A)),groups1707563613775114915nt_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8228_prod__dvd__prod__subset,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups705719431365010083at_int(F,A)),groups705719431365010083at_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8229_prod__dvd__prod__subset,axiom,
    ! [B: set_int,A: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A)),groups1705073143266064639nt_int(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8230_prod__dvd__prod__subset,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups708209901874060359at_nat(F,A)),groups708209901874060359at_nat(F,B))) ) ) ).

% prod_dvd_prod_subset
tff(fact_8231_le__floor__iff,axiom,
    ! [Z: int,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X2)) ) ).

% le_floor_iff
tff(fact_8232_le__floor__iff,axiom,
    ! [Z: int,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X2)) ) ).

% le_floor_iff
tff(fact_8233_floor__less__iff,axiom,
    ! [X2: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X2)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,Z))) ) ).

% floor_less_iff
tff(fact_8234_floor__less__iff,axiom,
    ! [X2: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X2)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% floor_less_iff
tff(fact_8235_int__add__floor,axiom,
    ! [Z: int,X2: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X2)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),X2)) ).

% int_add_floor
tff(fact_8236_int__add__floor,axiom,
    ! [Z: int,X2: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X2)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),X2)) ).

% int_add_floor
tff(fact_8237_floor__add__int,axiom,
    ! [X2: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_8238_floor__add__int,axiom,
    ! [X2: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X2),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_8239_le__floor__add,axiom,
    ! [X2: real,Y3: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim6058952711729229775r_real,Y3))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)))) ).

% le_floor_add
tff(fact_8240_le__floor__add,axiom,
    ! [X2: rat,Y3: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim3151403230148437115or_rat,Y3))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X2),Y3)))) ).

% le_floor_add
tff(fact_8241_floor__power,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X2)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_real_int(archim6058952711729229775r_real,X2)),N) ) ) ).

% floor_power
tff(fact_8242_floor__power,axiom,
    ! [X2: rat,N: nat] :
      ( ( X2 = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X2)) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(X2),N)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_rat_int(archim3151403230148437115or_rat,X2)),N) ) ) ).

% floor_power
tff(fact_8243_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,K)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_8244_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_8245_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,N))),groups129246275422532515t_real(G,set_or4665077453230672383an_nat(N,P2))) = groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8246_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))),groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(N,P2))) = groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8247_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,N))),groups705719431365010083at_int(G,set_or4665077453230672383an_nat(N,P2))) = groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8248_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,N))),groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(N,P2))) = groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8249_gbinomial__altdef__of__nat,axiom,
    ! [A2: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),K) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_aav(A2),K),set_or4665077453230672383an_nat(zero_zero_nat,K)) ).

% gbinomial_altdef_of_nat
tff(fact_8250_gbinomial__altdef__of__nat,axiom,
    ! [A2: rat,K: nat] : aa_nat_rat(gbinomial_rat(A2),K) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_aaw(A2),K),set_or4665077453230672383an_nat(zero_zero_nat,K)) ).

% gbinomial_altdef_of_nat
tff(fact_8251_gbinomial__altdef__of__nat,axiom,
    ! [A2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_aax(A2),K),set_or4665077453230672383an_nat(zero_zero_nat,K)) ).

% gbinomial_altdef_of_nat
tff(fact_8252_gbinomial__Suc__Suc,axiom,
    ! [A2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8253_gbinomial__Suc__Suc,axiom,
    ! [A2: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(A2),K)),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8254_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8255_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_assn] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups7882442080178216443t_assn(G,minus_minus_set_int(A,collect_int(aTP_Lamm_aay(G)))) = groups7882442080178216443t_assn(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8256_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_aaz(G)))) = groups1304777262505850412r_assn(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8257_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups4150731942483176573x_assn(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_aba(G)))) = groups4150731942483176573x_assn(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8258_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A,collect_int(aTP_Lamm_abb(G)))) = groups2316167850115554303t_real(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8259_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_abc(G)))) = groups9004974159866482096r_real(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8260_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_abd(G)))) = groups766887009212190081x_real(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8261_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A,collect_int(aTP_Lamm_abe(G)))) = groups1072433553688619179nt_rat(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8262_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,collect_Code_integer(aTP_Lamm_abf(G)))) = groups2555765274223993564er_rat(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8263_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A,collect_complex(aTP_Lamm_abg(G)))) = groups225925009352817453ex_rat(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8264_prod_Osetdiff__irrelevant,axiom,
    ! [A: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A,collect_int(aTP_Lamm_abh(G)))) = groups1707563613775114915nt_nat(G,A) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8265_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abi(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_8266_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sh(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_8267_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(N,M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abj(G),N),M),set_or1269000886237332187st_nat(N,M)) ).

% prod.atLeastAtMost_rev
tff(fact_8268_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(N,M)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(G),N),M),set_or1269000886237332187st_nat(N,M)) ).

% prod.atLeastAtMost_rev
tff(fact_8269_less__1__prod2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I)))
         => ( ! [I2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8270_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8271_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups129246275422532515t_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8272_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8273_less__1__prod2,axiom,
    ! [I6: set_Code_integer,I: code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Code_integer_real(F,I)))
         => ( ! [I2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Code_integer_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups9004974159866482096r_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8274_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8275_less__1__prod2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I2: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8276_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I)))
         => ( ! [I2: real] :
                ( pp(aa_set_real_bool(member_real(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8277_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8278_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_8279_less__1__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8280_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8281_less__1__prod,axiom,
    ! [I6: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( ( I6 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Code_integer_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups9004974159866482096r_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8282_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8283_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups129246275422532515t_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8284_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8285_less__1__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8286_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( pp(aa_set_real_bool(member_real(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8287_less__1__prod,axiom,
    ! [I6: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I6))
     => ( ( I6 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_Code_integer_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8288_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_8289_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_assn(G,X4) = one_one_assn ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_assn(G,X4) = aa_VEBT_VEBT_assn(H,X4) ) )
           => ( groups569574905686396791T_assn(G,T5) = groups569574905686396791T_assn(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8290_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_assn(G,X4) = one_one_assn ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_assn(G,X4) = aa_real_assn(H,X4) ) )
           => ( groups1155561341820557179l_assn(G,T5) = groups1155561341820557179l_assn(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8291_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn,H: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_assn(G,X4) = one_one_assn ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_assn(G,X4) = aa_Code_integer_assn(H,X4) ) )
           => ( groups1304777262505850412r_assn(G,T5) = groups1304777262505850412r_assn(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8292_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_assn,H: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_assn(G,X4) = one_one_assn ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ( aa_complex_assn(G,X4) = aa_complex_assn(H,X4) ) )
           => ( groups4150731942483176573x_assn(G,T5) = groups4150731942483176573x_assn(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8293_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_real(G,X4) = one_one_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2703838992350267259T_real(G,T5) = groups2703838992350267259T_real(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8294_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_real(G,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,T5) = groups1681761925125756287l_real(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8295_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(G,X4) = one_one_real ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_real(G,X4) = aa_Code_integer_real(H,X4) ) )
           => ( groups9004974159866482096r_real(G,T5) = groups9004974159866482096r_real(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8296_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,T5) = groups766887009212190081x_real(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8297_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups5726676334696518183BT_rat(G,T5) = groups5726676334696518183BT_rat(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8298_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups4061424788464935467al_rat(G,T5) = groups4061424788464935467al_rat(H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8299_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,H: fun_VEBT_VEBT_assn,G: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_assn(H,X4) = one_one_assn ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_assn(G,X4) = aa_VEBT_VEBT_assn(H,X4) ) )
           => ( groups569574905686396791T_assn(G,S) = groups569574905686396791T_assn(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8300_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S: set_real,H: fun_real_assn,G: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_assn(H,X4) = one_one_assn ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_assn(G,X4) = aa_real_assn(H,X4) ) )
           => ( groups1155561341820557179l_assn(G,S) = groups1155561341820557179l_assn(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8301_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,H: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_assn(H,X4) = one_one_assn ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_assn(G,X4) = aa_Code_integer_assn(H,X4) ) )
           => ( groups1304777262505850412r_assn(G,S) = groups1304777262505850412r_assn(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8302_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S: set_complex,H: fun_complex_assn,G: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_assn(H,X4) = one_one_assn ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ( aa_complex_assn(G,X4) = aa_complex_assn(H,X4) ) )
           => ( groups4150731942483176573x_assn(G,S) = groups4150731942483176573x_assn(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8303_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_real(H,X4) = one_one_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2703838992350267259T_real(G,S) = groups2703838992350267259T_real(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8304_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_real(H,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,S) = groups1681761925125756287l_real(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8305_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(H,X4) = one_one_real ) )
         => ( ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),S))
               => ( aa_Code_integer_real(G,X4) = aa_Code_integer_real(H,X4) ) )
           => ( groups9004974159866482096r_real(G,S) = groups9004974159866482096r_real(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8306_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(H,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,S) = groups766887009212190081x_real(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8307_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S),T5))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(T5,S)))
             => ( aa_VEBT_VEBT_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),S))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups5726676334696518183BT_rat(G,S) = groups5726676334696518183BT_rat(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8308_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S),T5))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T5,S)))
             => ( aa_real_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups4061424788464935467al_rat(G,S) = groups4061424788464935467al_rat(H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8309_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_assn(G,X4) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,T5) = groups1304777262505850412r_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8310_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_assn(G,X4) = one_one_assn ) )
         => ( groups4150731942483176573x_assn(G,T5) = groups4150731942483176573x_assn(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8311_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(G,X4) = one_one_real ) )
         => ( groups9004974159866482096r_real(G,T5) = groups9004974159866482096r_real(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8312_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T5) = groups766887009212190081x_real(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8313_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_rat(G,X4) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,T5) = groups2555765274223993564er_rat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8314_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T5) = groups225925009352817453ex_rat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8315_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_nat(G,X4) = one_one_nat ) )
         => ( groups3190895334310489300er_nat(G,T5) = groups3190895334310489300er_nat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8316_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T5) = groups861055069439313189ex_nat(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8317_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_int(G,X4) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,T5) = groups3188404863801439024er_int(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8318_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T5) = groups858564598930262913ex_int(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8319_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_assn(G,X4) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,S) = groups1304777262505850412r_assn(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8320_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_assn(G,X4) = one_one_assn ) )
         => ( groups4150731942483176573x_assn(G,S) = groups4150731942483176573x_assn(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8321_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_real(G,X4) = one_one_real ) )
         => ( groups9004974159866482096r_real(G,S) = groups9004974159866482096r_real(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8322_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S) = groups766887009212190081x_real(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8323_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_rat(G,X4) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,S) = groups2555765274223993564er_rat(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8324_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S) = groups225925009352817453ex_rat(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8325_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_nat(G,X4) = one_one_nat ) )
         => ( groups3190895334310489300er_nat(G,S) = groups3190895334310489300er_nat(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8326_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S) = groups861055069439313189ex_nat(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8327_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T5))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S),T5))
       => ( ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),minus_2355218937544613996nteger(T5,S)))
             => ( aa_Code_integer_int(G,X4) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,S) = groups3188404863801439024er_int(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8328_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S),T5))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T5,S)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S) = groups858564598930262913ex_int(G,T5) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8329_prod_Osame__carrierI,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_assn(H,B3) = one_one_assn ) )
             => ( ( groups569574905686396791T_assn(G,C) = groups569574905686396791T_assn(H,C) )
               => ( groups569574905686396791T_assn(G,A) = groups569574905686396791T_assn(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8330_prod_Osame__carrierI,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_assn(H,B3) = one_one_assn ) )
             => ( ( groups1155561341820557179l_assn(G,C) = groups1155561341820557179l_assn(H,C) )
               => ( groups1155561341820557179l_assn(G,A) = groups1155561341820557179l_assn(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8331_prod_Osame__carrierI,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co4490034186337852034r_assn,H: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_assn(H,B3) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(G,C) = groups1304777262505850412r_assn(H,C) )
               => ( groups1304777262505850412r_assn(G,A) = groups1304777262505850412r_assn(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8332_prod_Osame__carrierI,axiom,
    ! [C: set_complex,A: set_complex,B: set_complex,G: fun_complex_assn,H: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),C))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C,A)))
               => ( aa_complex_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C,B)))
                 => ( aa_complex_assn(H,B3) = one_one_assn ) )
             => ( ( groups4150731942483176573x_assn(G,C) = groups4150731942483176573x_assn(H,C) )
               => ( groups4150731942483176573x_assn(G,A) = groups4150731942483176573x_assn(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8333_prod_Osame__carrierI,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_real(H,B3) = one_one_real ) )
             => ( ( groups2703838992350267259T_real(G,C) = groups2703838992350267259T_real(H,C) )
               => ( groups2703838992350267259T_real(G,A) = groups2703838992350267259T_real(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8334_prod_Osame__carrierI,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C) = groups1681761925125756287l_real(H,C) )
               => ( groups1681761925125756287l_real(G,A) = groups1681761925125756287l_real(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8335_prod_Osame__carrierI,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_real(G,A3) = one_one_real ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_real(H,B3) = one_one_real ) )
             => ( ( groups9004974159866482096r_real(G,C) = groups9004974159866482096r_real(H,C) )
               => ( groups9004974159866482096r_real(G,A) = groups9004974159866482096r_real(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8336_prod_Osame__carrierI,axiom,
    ! [C: set_complex,A: set_complex,B: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),C))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C,A)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C,B)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C) = groups766887009212190081x_real(H,C) )
               => ( groups766887009212190081x_real(G,A) = groups766887009212190081x_real(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8337_prod_Osame__carrierI,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = one_one_rat ) )
             => ( ( groups5726676334696518183BT_rat(G,C) = groups5726676334696518183BT_rat(H,C) )
               => ( groups5726676334696518183BT_rat(G,A) = groups5726676334696518183BT_rat(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8338_prod_Osame__carrierI,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C) = groups4061424788464935467al_rat(H,C) )
               => ( groups4061424788464935467al_rat(G,A) = groups4061424788464935467al_rat(H,B) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8339_prod_Osame__carrier,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_assn(H,B3) = one_one_assn ) )
             => ( ( groups569574905686396791T_assn(G,A) = groups569574905686396791T_assn(H,B) )
              <=> ( groups569574905686396791T_assn(G,C) = groups569574905686396791T_assn(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8340_prod_Osame__carrier,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_assn(H,B3) = one_one_assn ) )
             => ( ( groups1155561341820557179l_assn(G,A) = groups1155561341820557179l_assn(H,B) )
              <=> ( groups1155561341820557179l_assn(G,C) = groups1155561341820557179l_assn(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8341_prod_Osame__carrier,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co4490034186337852034r_assn,H: fun_Co4490034186337852034r_assn] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_assn(H,B3) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(G,A) = groups1304777262505850412r_assn(H,B) )
              <=> ( groups1304777262505850412r_assn(G,C) = groups1304777262505850412r_assn(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8342_prod_Osame__carrier,axiom,
    ! [C: set_complex,A: set_complex,B: set_complex,G: fun_complex_assn,H: fun_complex_assn] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),C))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C,A)))
               => ( aa_complex_assn(G,A3) = one_one_assn ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C,B)))
                 => ( aa_complex_assn(H,B3) = one_one_assn ) )
             => ( ( groups4150731942483176573x_assn(G,A) = groups4150731942483176573x_assn(H,B) )
              <=> ( groups4150731942483176573x_assn(G,C) = groups4150731942483176573x_assn(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8343_prod_Osame__carrier,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_real(H,B3) = one_one_real ) )
             => ( ( groups2703838992350267259T_real(G,A) = groups2703838992350267259T_real(H,B) )
              <=> ( groups2703838992350267259T_real(G,C) = groups2703838992350267259T_real(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8344_prod_Osame__carrier,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A) = groups1681761925125756287l_real(H,B) )
              <=> ( groups1681761925125756287l_real(G,C) = groups1681761925125756287l_real(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8345_prod_Osame__carrier,axiom,
    ! [C: set_Code_integer,A: set_Code_integer,B: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),C))
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),C))
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),minus_2355218937544613996nteger(C,A)))
               => ( aa_Code_integer_real(G,A3) = one_one_real ) )
           => ( ! [B3: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(C,B)))
                 => ( aa_Code_integer_real(H,B3) = one_one_real ) )
             => ( ( groups9004974159866482096r_real(G,A) = groups9004974159866482096r_real(H,B) )
              <=> ( groups9004974159866482096r_real(G,C) = groups9004974159866482096r_real(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8346_prod_Osame__carrier,axiom,
    ! [C: set_complex,A: set_complex,B: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),C))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),C))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C,A)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C,B)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A) = groups766887009212190081x_real(H,B) )
              <=> ( groups766887009212190081x_real(G,C) = groups766887009212190081x_real(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8347_prod_Osame__carrier,axiom,
    ! [C: set_VEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),C))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B),C))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C,A)))
               => ( aa_VEBT_VEBT_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C,B)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = one_one_rat ) )
             => ( ( groups5726676334696518183BT_rat(G,A) = groups5726676334696518183BT_rat(H,B) )
              <=> ( groups5726676334696518183BT_rat(G,C) = groups5726676334696518183BT_rat(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8348_prod_Osame__carrier,axiom,
    ! [C: set_real,A: set_real,B: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),C))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B),C))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C,A)))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C,B)))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A) = groups4061424788464935467al_rat(H,B) )
              <=> ( groups4061424788464935467al_rat(G,C) = groups4061424788464935467al_rat(H,C) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8349_prod_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups9004974159866482096r_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A,B))),groups9004974159866482096r_real(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8350_prod_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups766887009212190081x_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A,B))),groups766887009212190081x_real(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8351_prod_Osubset__diff,axiom,
    ! [B: set_nat,A: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A))
       => ( groups129246275422532515t_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,minus_minus_set_nat(A,B))),groups129246275422532515t_real(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8352_prod_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups2555765274223993564er_rat(G,A) = aa_rat_rat(times_times_rat(groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,B))),groups2555765274223993564er_rat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8353_prod_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups225925009352817453ex_rat(G,A) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A,B))),groups225925009352817453ex_rat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8354_prod_Osubset__diff,axiom,
    ! [B: set_nat,A: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A))
       => ( groups73079841787564623at_rat(G,A) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A,B))),groups73079841787564623at_rat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8355_prod_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups3190895334310489300er_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(G,minus_2355218937544613996nteger(A,B))),groups3190895334310489300er_nat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8356_prod_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups861055069439313189ex_nat(G,A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A,B))),groups861055069439313189ex_nat(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8357_prod_Osubset__diff,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B),A))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
       => ( groups3188404863801439024er_int(G,A) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,minus_2355218937544613996nteger(A,B))),groups3188404863801439024er_int(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8358_prod_Osubset__diff,axiom,
    ! [B: set_complex,A: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B),A))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
       => ( groups858564598930262913ex_int(G,A) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A,B))),groups858564598930262913ex_int(G,B)) ) ) ) ).

% prod.subset_diff
tff(fact_8359_of__nat__floor,axiom,
    ! [R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2)) ) ).

% of_nat_floor
tff(fact_8360_of__nat__floor,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R2)))),R2)) ) ).

% of_nat_floor
tff(fact_8361_one__add__floor,axiom,
    ! [X2: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),one_one_real)) ).

% one_add_floor
tff(fact_8362_one__add__floor,axiom,
    ! [X2: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X2),one_one_rat)) ).

% one_add_floor
tff(fact_8363_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% floor_divide_of_nat_eq
tff(fact_8364_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% floor_divide_of_nat_eq
tff(fact_8365_le__mult__nat__floor,axiom,
    ! [A2: real,B2: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A2))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))))) ).

% le_mult_nat_floor
tff(fact_8366_le__mult__nat__floor,axiom,
    ! [A2: rat,B2: rat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,A2))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A2),B2))))) ).

% le_mult_nat_floor
tff(fact_8367_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_real(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8368_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_rat(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8369_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_int(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8370_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_nat_nat(G,N)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8371_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8372_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8373_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8374_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8375_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8376_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8377_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8378_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8379_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups129246275422532515t_real(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_real(G,B2)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8380_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_rat(G,B2)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8381_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups705719431365010083at_int(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_int(G,B2)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8382_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(A2,aa_nat_nat(suc,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(A2,B2))),aa_nat_nat(G,B2)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8383_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8384_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8385_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8386_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8387_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8388_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8389_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8390_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8391_nat__floor__neg,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X2)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_8392_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,N)),groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8393_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,N)),groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8394_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,N)),groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8395_prod_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,N)),groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,N))) ) ) ).

% prod.last_plus
tff(fact_8396_floor__eq3,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X2)) = N ) ) ) ).

% floor_eq3
tff(fact_8397_ceiling__altdef,axiom,
    ! [X2: real] :
      ( ( ( X2 = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X2)) )
       => ( aa_real_int(archim7802044766580827645g_real,X2) = aa_real_int(archim6058952711729229775r_real,X2) ) )
      & ( ( X2 != aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X2)) )
       => ( aa_real_int(archim7802044766580827645g_real,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),one_one_int) ) ) ) ).

% ceiling_altdef
tff(fact_8398_ceiling__altdef,axiom,
    ! [X2: rat] :
      ( ( ( X2 = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X2)) )
       => ( aa_rat_int(archim2889992004027027881ng_rat,X2) = aa_rat_int(archim3151403230148437115or_rat,X2) ) )
      & ( ( X2 != aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X2)) )
       => ( aa_rat_int(archim2889992004027027881ng_rat,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),one_one_int) ) ) ) ).

% ceiling_altdef
tff(fact_8399_le__nat__floor,axiom,
    ! [X2: nat,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X2)),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A2)))) ) ).

% le_nat_floor
tff(fact_8400_gbinomial__addition__formula,axiom,
    ! [A2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8401_gbinomial__addition__formula,axiom,
    ! [A2: rat,K: nat] : aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_8402_ceiling__diff__floor__le__1,axiom,
    ! [X2: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X2)),aa_real_int(archim6058952711729229775r_real,X2))),one_one_int)) ).

% ceiling_diff_floor_le_1
tff(fact_8403_ceiling__diff__floor__le__1,axiom,
    ! [X2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X2)),aa_rat_int(archim3151403230148437115or_rat,X2))),one_one_int)) ).

% ceiling_diff_floor_le_1
tff(fact_8404_gbinomial__mult__1,axiom,
    ! [A2: rat,K: nat] : aa_rat_rat(times_times_rat(A2),aa_nat_rat(gbinomial_rat(A2),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A2),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8405_gbinomial__mult__1,axiom,
    ! [A2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8406_gbinomial__mult__1_H,axiom,
    ! [A2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A2),K)),A2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A2),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8407_gbinomial__mult__1_H,axiom,
    ! [A2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8408_gbinomial__absorb__comp,axiom,
    ! [A2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A2),K)) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_8409_gbinomial__absorb__comp,axiom,
    ! [A2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_8410_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8411_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A2),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8412_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_8413_floor__eq,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X2) = N ) ) ) ).

% floor_eq
tff(fact_8414_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_8415_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8416_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_8417_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8418_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_qk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8419_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8420_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_pp(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8421_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(aTP_Lamm_pq(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8422_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_qk(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8423_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_ql(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8424_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),groups708209901874060359at_nat(aTP_Lamm_pp(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8425_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : groups705719431365010083at_int(G,set_or4665077453230672383an_nat(N,M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abk(G),N),M),set_or4665077453230672383an_nat(N,M)) ).

% prod.atLeastLessThan_rev
tff(fact_8426_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(N,M)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qp(G),N),M),set_or4665077453230672383an_nat(N,M)) ).

% prod.atLeastLessThan_rev
tff(fact_8427_prod_Onested__swap,axiom,
    ! [A2: fun_nat_fun_nat_int,N: nat] : groups705719431365010083at_int(aTP_Lamm_abl(A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abm(A2),N),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% prod.nested_swap
tff(fact_8428_prod_Onested__swap,axiom,
    ! [A2: fun_nat_fun_nat_nat,N: nat] : groups708209901874060359at_nat(aTP_Lamm_abn(A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abo(A2),N),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% prod.nested_swap
tff(fact_8429_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = groups705719431365010083at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_8430_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = groups708209901874060359at_nat(aTP_Lamm_pp(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_8431_prod_Onat__group,axiom,
    ! [G: fun_nat_int,K: nat,N: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abp(G),K),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% prod.nat_group
tff(fact_8432_prod_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abq(G),K),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% prod.nat_group
tff(fact_8433_prod__mono__strict,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_VEBT_VEBT_real(F,I2)),aa_VEBT_VEBT_real(G,I2))) ) )
       => ( ( A != bot_bo8194388402131092736T_VEBT )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2703838992350267259T_real(F,A)),groups2703838992350267259T_real(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8434_prod__mono__strict,axiom,
    ! [A: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),A))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,I2)),aa_real_real(G,I2))) ) )
       => ( ( A != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F,A)),groups1681761925125756287l_real(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8435_prod__mono__strict,axiom,
    ! [A: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),A))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Code_integer_real(F,I2)),aa_Code_integer_real(G,I2))) ) )
       => ( ( A != bot_bo3990330152332043303nteger )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups9004974159866482096r_real(F,A)),groups9004974159866482096r_real(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8436_prod__mono__strict,axiom,
    ! [A: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),A))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,I2)),aa_complex_real(G,I2))) ) )
       => ( ( A != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F,A)),groups766887009212190081x_real(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8437_prod__mono__strict,axiom,
    ! [A: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),A))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,I2)),aa_nat_real(G,I2))) ) )
       => ( ( A != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups129246275422532515t_real(F,A)),groups129246275422532515t_real(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8438_prod__mono__strict,axiom,
    ! [A: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),A))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,I2)),aa_int_real(G,I2))) ) )
       => ( ( A != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F,A)),groups2316167850115554303t_real(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8439_prod__mono__strict,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2))) ) )
       => ( ( A != bot_bo8194388402131092736T_VEBT )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5726676334696518183BT_rat(F,A)),groups5726676334696518183BT_rat(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8440_prod__mono__strict,axiom,
    ! [A: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),A))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) ) )
       => ( ( A != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups4061424788464935467al_rat(F,A)),groups4061424788464935467al_rat(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8441_prod__mono__strict,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ! [I2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),A))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,I2)),aa_Code_integer_rat(G,I2))) ) )
       => ( ( A != bot_bo3990330152332043303nteger )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2555765274223993564er_rat(F,A)),groups2555765274223993564er_rat(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8442_prod__mono__strict,axiom,
    ! [A: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),A))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) ) )
       => ( ( A != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups225925009352817453ex_rat(F,A)),groups225925009352817453ex_rat(G,A))) ) ) ) ).

% prod_mono_strict
tff(fact_8443_even__prod__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups1707563613775114915nt_nat(F,A)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8444_even__prod__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups3190895334310489300er_nat(F,A)))
      <=> ? [X3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_Code_integer_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8445_even__prod__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8446_even__prod__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3188404863801439024er_int(F,A)))
      <=> ? [X3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8447_even__prod__iff,axiom,
    ! [A: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8448_even__prod__iff,axiom,
    ! [A: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8449_even__prod__iff,axiom,
    ! [A: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups1705073143266064639nt_int(F,A)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8450_even__prod__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups708209901874060359at_nat(F,A)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8451_floor__unique,axiom,
    ! [Z: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X2) = Z ) ) ) ).

% floor_unique
tff(fact_8452_floor__unique,axiom,
    ! [Z: int,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)))
       => ( aa_rat_int(archim3151403230148437115or_rat,X2) = Z ) ) ) ).

% floor_unique
tff(fact_8453_floor__eq__iff,axiom,
    ! [X2: real,A2: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X2) = A2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),X2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A2)),one_one_real))) ) ) ).

% floor_eq_iff
tff(fact_8454_floor__eq__iff,axiom,
    ! [X2: rat,A2: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X2) = A2 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),X2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A2)),one_one_rat))) ) ) ).

% floor_eq_iff
tff(fact_8455_floor__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim6058952711729229775r_real,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% floor_split
tff(fact_8456_floor__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim3151403230148437115or_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% floor_split
tff(fact_8457_prod_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% prod.insert_remove
tff(fact_8458_prod_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Co3960896333048504454r_real,X2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups9004974159866482096r_real(G,insert_Code_integer(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,X2)),groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.insert_remove
tff(fact_8459_prod_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_real,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups766887009212190081x_real(G,insert_complex(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_8460_prod_Oinsert__remove,axiom,
    ! [A: set_int,G: fun_int_real,X2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups2316167850115554303t_real(G,insert_int(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,minus_minus_set_int(A,insert_int(X2,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_8461_prod_Oinsert__remove,axiom,
    ! [A: set_nat,G: fun_nat_real,X2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups129246275422532515t_real(G,insert_nat(X2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),groups129246275422532515t_real(G,minus_minus_set_nat(A,insert_nat(X2,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_8462_prod_Oinsert__remove,axiom,
    ! [A: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% prod.insert_remove
tff(fact_8463_prod_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,X2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( groups2555765274223993564er_rat(G,insert_Code_integer(X2,A)) = aa_rat_rat(times_times_rat(aa_Code_integer_rat(G,X2)),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.insert_remove
tff(fact_8464_prod_Oinsert__remove,axiom,
    ! [A: set_complex,G: fun_complex_rat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( groups225925009352817453ex_rat(G,insert_complex(X2,A)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X2)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_8465_prod_Oinsert__remove,axiom,
    ! [A: set_int,G: fun_int_rat,X2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( groups1072433553688619179nt_rat(G,insert_int(X2,A)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A,insert_int(X2,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_8466_prod_Oinsert__remove,axiom,
    ! [A: set_nat,G: fun_nat_rat,X2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( groups73079841787564623at_rat(G,insert_nat(X2,A)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,minus_minus_set_nat(A,insert_nat(X2,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_8467_prod_Oremove,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups2703838992350267259T_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% prod.remove
tff(fact_8468_prod_Oremove,axiom,
    ! [A: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => ( groups1681761925125756287l_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,minus_minus_set_real(A,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_8469_prod_Oremove,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups9004974159866482096r_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,X2)),groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ) ).

% prod.remove
tff(fact_8470_prod_Oremove,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(member_complex(X2),A))
       => ( groups766887009212190081x_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_8471_prod_Oremove,axiom,
    ! [A: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_set_int_bool(member_int(X2),A))
       => ( groups2316167850115554303t_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,minus_minus_set_int(A,insert_int(X2,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_8472_prod_Oremove,axiom,
    ! [A: set_nat,X2: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_set_nat_bool(member_nat(X2),A))
       => ( groups129246275422532515t_real(G,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),groups129246275422532515t_real(G,minus_minus_set_nat(A,insert_nat(X2,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_8473_prod_Oremove,axiom,
    ! [A: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X2),A))
       => ( groups5726676334696518183BT_rat(G,A) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% prod.remove
tff(fact_8474_prod_Oremove,axiom,
    ! [A: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( pp(aa_set_real_bool(member_real(X2),A))
       => ( groups4061424788464935467al_rat(G,A) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,minus_minus_set_real(A,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_8475_prod_Oremove,axiom,
    ! [A: set_Code_integer,X2: code_integer,G: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),A))
       => ( groups2555765274223993564er_rat(G,A) = aa_rat_rat(times_times_rat(aa_Code_integer_rat(G,X2)),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,insert_Code_integer(X2,bot_bo3990330152332043303nteger)))) ) ) ) ).

% prod.remove
tff(fact_8476_prod_Oremove,axiom,
    ! [A: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_set_complex_bool(member_complex(X2),A))
       => ( groups225925009352817453ex_rat(G,A) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X2)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_8477_less__floor__iff,axiom,
    ! [Z: int,X2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X2)) ) ).

% less_floor_iff
tff(fact_8478_less__floor__iff,axiom,
    ! [Z: int,X2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X2)) ) ).

% less_floor_iff
tff(fact_8479_floor__le__iff,axiom,
    ! [X2: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X2)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% floor_le_iff
tff(fact_8480_floor__le__iff,axiom,
    ! [X2: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X2)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% floor_le_iff
tff(fact_8481_le__mult__floor,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A2)),aa_real_int(archim6058952711729229775r_real,B2))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)))) ) ) ).

% le_mult_floor
tff(fact_8482_le__mult__floor,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A2)),aa_rat_int(archim3151403230148437115or_rat,B2))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A2),B2)))) ) ) ).

% le_mult_floor
tff(fact_8483_floor__correct,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X2))),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),one_one_int)))) ) ).

% floor_correct
tff(fact_8484_floor__correct,axiom,
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X2))),X2))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),one_one_int)))) ) ).

% floor_correct
tff(fact_8485_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N))),groups129246275422532515t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8486_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8487_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8488_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N))),groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8489_prod_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_assn] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6906906614972039071t_assn(G,set_or1269000886237332187st_nat(M,N)) = one_one_assn ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6906906614972039071t_assn(G,set_or1269000886237332187st_nat(M,N)) = aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,set_or4665077453230672383an_nat(M,N))),aa_nat_assn(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8490_prod_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N)) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups129246275422532515t_real(G,set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or4665077453230672383an_nat(M,N))),aa_nat_real(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8491_prod_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N)) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_rat(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8492_prod_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N)) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or4665077453230672383an_nat(M,N))),aa_nat_int(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8493_prod_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N)) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(M,N))),aa_nat_nat(G,N)) ) ) ) ).

% prod.head_if
tff(fact_8494_floor__eq4,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X2)) = N ) ) ) ).

% floor_eq4
tff(fact_8495_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X2: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X2,Xa),Xb),Xc) = Y3 )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
         => ( Y3 = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
         => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X2,Xa),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_8496_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B2: nat,A2: nat,F: fun_nat_fun_nat_nat,Acc2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A2),B2),Acc2) = Acc2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A2),B2),Acc2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F,A2),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_8497_Suc__times__gbinomial,axiom,
    ! [K: nat,A2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_rat(gbinomial_rat(A2),K)) ).

% Suc_times_gbinomial
tff(fact_8498_Suc__times__gbinomial,axiom,
    ! [K: nat,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)) ).

% Suc_times_gbinomial
tff(fact_8499_gbinomial__absorption,axiom,
    ! [K: nat,A2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8500_gbinomial__absorption,axiom,
    ! [K: nat,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_8501_floor__eq2,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X2) = N ) ) ) ).

% floor_eq2
tff(fact_8502_prod_Odelta__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real,C2: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_qa(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(B2,A2)),groups2703838992350267259T_real(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_qa(A2),B2),C2),S) = groups2703838992350267259T_real(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8503_prod_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B2: fun_real_real,C2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qb(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B2,A2)),groups1681761925125756287l_real(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qb(A2),B2),C2),S) = groups1681761925125756287l_real(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8504_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real,C2: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups9004974159866482096r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_qc(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(B2,A2)),groups9004974159866482096r_real(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups9004974159866482096r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_qc(A2),B2),C2),S) = groups9004974159866482096r_real(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8505_prod_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_real,C2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_qd(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B2,A2)),groups766887009212190081x_real(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_qd(A2),B2),C2),S) = groups766887009212190081x_real(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8506_prod_Odelta__remove,axiom,
    ! [S: set_int,A2: int,B2: fun_int_real,C2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ( pp(aa_set_int_bool(member_int(A2),S))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_abr(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(B2,A2)),groups2316167850115554303t_real(C2,minus_minus_set_int(S,insert_int(A2,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A2),S))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_abr(A2),B2),C2),S) = groups2316167850115554303t_real(C2,minus_minus_set_int(S,insert_int(A2,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8507_prod_Odelta__remove,axiom,
    ! [S: set_nat,A2: nat,B2: fun_nat_real,C2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ( pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_abs(A2),B2),C2),S) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(B2,A2)),groups129246275422532515t_real(C2,minus_minus_set_nat(S,insert_nat(A2,bot_bot_set_nat)))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A2),S))
         => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_abs(A2),B2),C2),S) = groups129246275422532515t_real(C2,minus_minus_set_nat(S,insert_nat(A2,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8508_prod_Odelta__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_rat,C2: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_qe(A2),B2),C2),S) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(B2,A2)),groups5726676334696518183BT_rat(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),S))
         => ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_qe(A2),B2),C2),S) = groups5726676334696518183BT_rat(C2,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8509_prod_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B2: fun_real_rat,C2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ( pp(aa_set_real_bool(member_real(A2),S))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_qf(A2),B2),C2),S) = aa_rat_rat(times_times_rat(aa_real_rat(B2,A2)),groups4061424788464935467al_rat(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A2),S))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_qf(A2),B2),C2),S) = groups4061424788464935467al_rat(C2,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8510_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer,B2: fun_Code_integer_rat,C2: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups2555765274223993564er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_qg(A2),B2),C2),S) = aa_rat_rat(times_times_rat(aa_Code_integer_rat(B2,A2)),groups2555765274223993564er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),S))
         => ( groups2555765274223993564er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_qg(A2),B2),C2),S) = groups2555765274223993564er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8511_prod_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B2: fun_complex_rat,C2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ( pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_qh(A2),B2),C2),S) = aa_rat_rat(times_times_rat(aa_complex_rat(B2,A2)),groups225925009352817453ex_rat(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A2),S))
         => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_qh(A2),B2),C2),S) = groups225925009352817453ex_rat(C2,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_8512_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_8513_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A2),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A2),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8514_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8515_norm__prod__diff,axiom,
    ! [I6: set_VEBT_VEBT,Z: fun_VEBT_VEBT_real,W: fun_VEBT_VEBT_real] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_VEBT_VEBT_real(Z,I2))),one_one_real)) )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_VEBT_VEBT_real(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2703838992350267259T_real(Z,I6)),groups2703838992350267259T_real(W,I6)))),groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_abt(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8516_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_real,W: fun_real_real] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(Z,I2))),one_one_real)) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1681761925125756287l_real(Z,I6)),groups1681761925125756287l_real(W,I6)))),groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_abu(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8517_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_real,W: fun_int_real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(Z,I2))),one_one_real)) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2316167850115554303t_real(Z,I6)),groups2316167850115554303t_real(W,I6)))),groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_abv(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8518_norm__prod__diff,axiom,
    ! [I6: set_VEBT_VEBT,Z: fun_VE5569070535936029429omplex,W: fun_VE5569070535936029429omplex] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_VEBT_VEBT_complex(Z,I2))),one_one_real)) )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_VEBT_VEBT_complex(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups127312072573709053omplex(Z,I6)),groups127312072573709053omplex(W,I6)))),groups2240296850493347238T_real(aa_fun9093735442950272378T_real(aTP_Lamm_abw(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8519_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_complex,W: fun_real_complex] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(Z,I2))),one_one_real)) )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups713298508707869441omplex(Z,I6)),groups713298508707869441omplex(W,I6)))),groups8097168146408367636l_real(aa_fun2786211375485122666l_real(aTP_Lamm_abx(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8520_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_complex,W: fun_int_complex] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(Z,I2))),one_one_real)) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7440179247065528705omplex(Z,I6)),groups7440179247065528705omplex(W,I6)))),groups8778361861064173332t_real(aa_fun1985759105522019690t_real(aTP_Lamm_aby(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8521_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_real,W: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Z,I2))),one_one_real)) )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups129246275422532515t_real(Z,I6)),groups129246275422532515t_real(W,I6)))),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_abz(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8522_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_complex,W: fun_nat_complex] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Z,I2))),one_one_real)) )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(W,I2))),one_one_real)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups6464643781859351333omplex(Z,I6)),groups6464643781859351333omplex(W,I6)))),groups6591440286371151544t_real(aa_fun7054779078796324018t_real(aTP_Lamm_aca(Z),W),I6))) ) ) ).

% norm_prod_diff
tff(fact_8523_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : groups705719431365010083at_int(G,set_or4665077453230672383an_nat(N,M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abj(G),N),M),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8524_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : groups708209901874060359at_nat(G,set_or4665077453230672383an_nat(N,M)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(G),N),M),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8525_pochhammer__prod,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),N) = groups73079841787564623at_rat(aTP_Lamm_acb(A2),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8526_pochhammer__prod,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_acc,A2),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8527_pochhammer__prod,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),N) = groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_acd,A2),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8528_pochhammer__prod,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),N) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ace,A2),set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% pochhammer_prod
tff(fact_8529_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),Q3)),P2)) ) ).

% floor_divide_lower
tff(fact_8530_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),Q3)),P2)) ) ).

% floor_divide_lower
tff(fact_8531_prod__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,B3))) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A)),groups2703838992350267259T_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8532_prod__mono2,axiom,
    ! [B: set_real,A: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A)),groups1681761925125756287l_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8533_prod__mono2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( ! [B3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Code_integer_real(F,B3))) )
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups9004974159866482096r_real(F,A)),groups9004974159866482096r_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8534_prod__mono2,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A)),groups766887009212190081x_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8535_prod__mono2,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,B3))) )
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),A))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A)),groups129246275422532515t_real(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8536_prod__mono2,axiom,
    ! [B: set_VEBT_VEBT,A: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A),B))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,B3))) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A)),groups5726676334696518183BT_rat(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8537_prod__mono2,axiom,
    ! [B: set_real,A: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A),B))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A)),groups4061424788464935467al_rat(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8538_prod__mono2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A),B))
       => ( ! [B3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B3),minus_2355218937544613996nteger(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(F,B3))) )
         => ( ! [A3: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2555765274223993564er_rat(F,A)),groups2555765274223993564er_rat(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8539_prod__mono2,axiom,
    ! [B: set_complex,A: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A),B))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A)),groups225925009352817453ex_rat(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8540_prod__mono2,axiom,
    ! [B: set_nat,A: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B,A)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B3))) )
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),A))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A)),groups73079841787564623at_rat(F,B))) ) ) ) ) ).

% prod_mono2
tff(fact_8541_prod__diff1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,A2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( aa_VEBT_VEBT_complex(F,A2) != zero_zero_complex )
       => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
           => ( groups127312072573709053omplex(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups127312072573709053omplex(F,A)),aa_VEBT_VEBT_complex(F,A2)) ) )
          & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
           => ( groups127312072573709053omplex(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = groups127312072573709053omplex(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8542_prod__diff1,axiom,
    ! [A: set_real,F: fun_real_complex,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( aa_real_complex(F,A2) != zero_zero_complex )
       => ( ( pp(aa_set_real_bool(member_real(A2),A))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups713298508707869441omplex(F,A)),aa_real_complex(F,A2)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A2),A))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = groups713298508707869441omplex(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8543_prod__diff1,axiom,
    ! [A: set_Code_integer,F: fun_Co1900716531650368264omplex,A2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( aa_Cod5621350732091595593omplex(F,A2) != zero_zero_complex )
       => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
           => ( groups862514429393162674omplex(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups862514429393162674omplex(F,A)),aa_Cod5621350732091595593omplex(F,A2)) ) )
          & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
           => ( groups862514429393162674omplex(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = groups862514429393162674omplex(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8544_prod__diff1,axiom,
    ! [A: set_complex,F: fun_complex_complex,A2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( aa_complex_complex(F,A2) != zero_zero_complex )
       => ( ( pp(aa_set_complex_bool(member_complex(A2),A))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups3708469109370488835omplex(F,A)),aa_complex_complex(F,A2)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A2),A))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = groups3708469109370488835omplex(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8545_prod__diff1,axiom,
    ! [A: set_int,F: fun_int_complex,A2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( aa_int_complex(F,A2) != zero_zero_complex )
       => ( ( pp(aa_set_int_bool(member_int(A2),A))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups7440179247065528705omplex(F,A)),aa_int_complex(F,A2)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A2),A))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = groups7440179247065528705omplex(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8546_prod__diff1,axiom,
    ! [A: set_nat,F: fun_nat_complex,A2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( aa_nat_complex(F,A2) != zero_zero_complex )
       => ( ( pp(aa_set_nat_bool(member_nat(A2),A))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups6464643781859351333omplex(F,A)),aa_nat_complex(F,A2)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A2),A))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = groups6464643781859351333omplex(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8547_prod__diff1,axiom,
    ! [A: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,A2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( aa_VEBT_VEBT_real(F,A2) != zero_zero_real )
       => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
           => ( groups2703838992350267259T_real(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups2703838992350267259T_real(F,A)),aa_VEBT_VEBT_real(F,A2)) ) )
          & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A2),A))
           => ( groups2703838992350267259T_real(F,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = groups2703838992350267259T_real(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8548_prod__diff1,axiom,
    ! [A: set_real,F: fun_real_real,A2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( aa_real_real(F,A2) != zero_zero_real )
       => ( ( pp(aa_set_real_bool(member_real(A2),A))
           => ( groups1681761925125756287l_real(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups1681761925125756287l_real(F,A)),aa_real_real(F,A2)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A2),A))
           => ( groups1681761925125756287l_real(F,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = groups1681761925125756287l_real(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8549_prod__diff1,axiom,
    ! [A: set_Code_integer,F: fun_Co3960896333048504454r_real,A2: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( aa_Code_integer_real(F,A2) != zero_zero_real )
       => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
           => ( groups9004974159866482096r_real(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups9004974159866482096r_real(F,A)),aa_Code_integer_real(F,A2)) ) )
          & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A2),A))
           => ( groups9004974159866482096r_real(F,minus_2355218937544613996nteger(A,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) = groups9004974159866482096r_real(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8550_prod__diff1,axiom,
    ! [A: set_complex,F: fun_complex_real,A2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( aa_complex_real(F,A2) != zero_zero_real )
       => ( ( pp(aa_set_complex_bool(member_complex(A2),A))
           => ( groups766887009212190081x_real(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups766887009212190081x_real(F,A)),aa_complex_real(F,A2)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A2),A))
           => ( groups766887009212190081x_real(F,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = groups766887009212190081x_real(F,A) ) ) ) ) ) ).

% prod_diff1
tff(fact_8551_gbinomial__factors,axiom,
    ! [A2: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),K)) ).

% gbinomial_factors
tff(fact_8552_gbinomial__factors,axiom,
    ! [A2: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A2),K)) ).

% gbinomial_factors
tff(fact_8553_gbinomial__factors,axiom,
    ! [A2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)) ).

% gbinomial_factors
tff(fact_8554_gbinomial__rec,axiom,
    ! [A2: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),K)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8555_gbinomial__rec,axiom,
    ! [A2: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A2),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8556_gbinomial__rec,axiom,
    ! [A2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8557_pochhammer__Suc__prod,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aTP_Lamm_acb(A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_8558_pochhammer__Suc__prod,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),aa_nat_nat(suc,N)) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_acc,A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_8559_pochhammer__Suc__prod,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_acd,A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_8560_pochhammer__Suc__prod,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),aa_nat_nat(suc,N)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ace,A2),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_8561_pochhammer__prod__rev,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),N) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_acf(A2),N),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_8562_pochhammer__prod__rev,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),N) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_acg(A2),N),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_8563_pochhammer__prod__rev,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ach(A2),N),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_8564_pochhammer__prod__rev,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),N) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aci(A2),N),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_8565_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),one_one_real)),Q3))) ) ).

% floor_divide_upper
tff(fact_8566_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),one_one_rat)),Q3))) ) ).

% floor_divide_upper
tff(fact_8567_round__def,axiom,
    ! [X2: real] : aa_real_int(archim8280529875227126926d_real,X2) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8568_round__def,axiom,
    ! [X2: rat] : aa_rat_int(archim7778729529865785530nd_rat,X2) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8569_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups129246275422532515t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups129246275422532515t_real(aTP_Lamm_acj(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_8570_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_ack(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_8571_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_acl(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_8572_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : groups708209901874060359at_nat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups708209901874060359at_nat(aTP_Lamm_acm(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_8573_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8574_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A2),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8575_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A2: nat,B2: nat] : groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(A2,B2)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_acn(F),A2),B2),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_8576_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A2: nat,B2: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(A2,B2)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_aco(F),A2,B2),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8577_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A2: nat,B2: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(A2,B2)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_acp(F),A2),B2),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8578_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A2: nat,B2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_acq(F),A2),B2),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8579_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A2: nat,B2: nat] : groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(A2,B2)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_acr(F),A2),B2),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8580_pochhammer__Suc__prod__rev,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A2),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_acf(A2),N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8581_pochhammer__Suc__prod__rev,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A2),aa_nat_nat(suc,N)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_acg(A2),N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8582_pochhammer__Suc__prod__rev,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A2),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ach(A2),N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8583_pochhammer__Suc__prod__rev,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A2),aa_nat_nat(suc,N)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aci(A2),N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8584_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_acs(K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8585_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_act,K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8586_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8587_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A2),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8588_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8589_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B2: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,K))),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8590_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8591_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8592_gbinomial__partial__row__sum,axiom,
    ! [A2: complex,M: nat] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_wy,A2),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8593_gbinomial__partial__row__sum,axiom,
    ! [A2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_wz(A2),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8594_gbinomial__partial__row__sum,axiom,
    ! [A2: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xa,A2),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8595_round__altdef,axiom,
    ! [X2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X2)))
       => ( aa_real_int(archim8280529875227126926d_real,X2) = aa_real_int(archim7802044766580827645g_real,X2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X2)))
       => ( aa_real_int(archim8280529875227126926d_real,X2) = aa_real_int(archim6058952711729229775r_real,X2) ) ) ) ).

% round_altdef
tff(fact_8596_round__altdef,axiom,
    ! [X2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X2)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X2) = aa_rat_int(archim2889992004027027881ng_rat,X2) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X2)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X2) = aa_rat_int(archim3151403230148437115or_rat,X2) ) ) ) ).

% round_altdef
tff(fact_8597_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,M))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8598_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8599_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),one_one_real)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8600_prod_Oinject,axiom,
    ! [X1: code_integer,X22: code_integer,Y15: code_integer,Y2: code_integer] :
      ( ( produc1086072967326762835nteger(X1,X22) = produc1086072967326762835nteger(Y15,Y2) )
    <=> ( ( X1 = Y15 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_8601_prod_Oinject,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X22: set_nat,Y15: heap_e7401611519738050253t_unit,Y2: set_nat] :
      ( ( produc7507926704131184380et_nat(X1,X22) = produc7507926704131184380et_nat(Y15,Y2) )
    <=> ( ( X1 = Y15 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_8602_prod_Oinject,axiom,
    ! [X1: num,X22: num,Y15: num,Y2: num] :
      ( ( product_Pair_num_num(X1,X22) = product_Pair_num_num(Y15,Y2) )
    <=> ( ( X1 = Y15 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_8603_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y15: nat,Y2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y15),Y2) )
    <=> ( ( X1 = Y15 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_8604_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y15: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(X1),X22) = aa_int7462565939943104719nt_int(product_Pair_int_int(Y15),Y2) )
    <=> ( ( X1 = Y15 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_8605_old_Oprod_Oinject,axiom,
    ! [A2: code_integer,B2: code_integer,A4: code_integer,B4: code_integer] :
      ( ( produc1086072967326762835nteger(A2,B2) = produc1086072967326762835nteger(A4,B4) )
    <=> ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_8606_old_Oprod_Oinject,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,A4: heap_e7401611519738050253t_unit,B4: set_nat] :
      ( ( produc7507926704131184380et_nat(A2,B2) = produc7507926704131184380et_nat(A4,B4) )
    <=> ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_8607_old_Oprod_Oinject,axiom,
    ! [A2: num,B2: num,A4: num,B4: num] :
      ( ( product_Pair_num_num(A2,B2) = product_Pair_num_num(A4,B4) )
    <=> ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_8608_old_Oprod_Oinject,axiom,
    ! [A2: nat,B2: nat,A4: nat,B4: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) )
    <=> ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_8609_old_Oprod_Oinject,axiom,
    ! [A2: int,B2: int,A4: int,B4: int] :
      ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2) = aa_int7462565939943104719nt_int(product_Pair_int_int(A4),B4) )
    <=> ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_8610_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_ord_atMost_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),K)) ) ).

% atMost_iff
tff(fact_8611_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( member_set_int(I,set_or58775011639299419et_int(K))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(I),K)) ) ).

% atMost_iff
tff(fact_8612_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_atMost_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),K)) ) ).

% atMost_iff
tff(fact_8613_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_atMost_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),K)) ) ).

% atMost_iff
tff(fact_8614_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_8615_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_atMost_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K)) ) ).

% atMost_iff
tff(fact_8616_atMost__subset__iff,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X2),set_or58775011639299419et_int(Y3))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y3)) ) ).

% atMost_subset_iff
tff(fact_8617_atMost__subset__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(X2)),set_ord_atMost_rat(Y3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3)) ) ).

% atMost_subset_iff
tff(fact_8618_atMost__subset__iff,axiom,
    ! [X2: num,Y3: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(X2)),set_ord_atMost_num(Y3)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y3)) ) ).

% atMost_subset_iff
tff(fact_8619_atMost__subset__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X2)),aa_nat_set_nat(set_ord_atMost_nat,Y3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y3)) ) ).

% atMost_subset_iff
tff(fact_8620_atMost__subset__iff,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X2)),set_ord_atMost_int(Y3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y3)) ) ).

% atMost_subset_iff
tff(fact_8621_prod__eq__1__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( groups1707563613775114915nt_nat(F,A) = one_one_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A))
           => ( aa_int_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8622_prod__eq__1__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( groups3190895334310489300er_nat(F,A) = one_one_nat )
      <=> ! [X3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
           => ( aa_Code_integer_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8623_prod__eq__1__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( groups861055069439313189ex_nat(F,A) = one_one_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A))
           => ( aa_complex_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8624_prod__eq__1__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( groups708209901874060359at_nat(F,A) = one_one_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A))
           => ( aa_nat_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8625_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H4: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H4))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
        | pp(aa_set_int_bool(ord_less_eq_set_int(H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8626_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H4: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(H4)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8627_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H4: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(H4)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8628_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H4: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_or1269000886237332187st_nat(L,H)),aa_nat_set_nat(set_ord_atMost_nat,H4)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8629_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H4: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H4)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8630_Icc__subset__Iic__iff,axiom,
    ! [L: code_integer,H: code_integer,H4: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or189985376899183464nteger(L,H)),set_or9101266186257409494nteger(H4)))
    <=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),H))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8631_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H4: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H4)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),H4)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8632_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8633_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8634_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8635_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_8636_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8637_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8638_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8639_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_8640_prod__pos__nat__iff,axiom,
    ! [A: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A)))
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8641_prod__pos__nat__iff,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups3190895334310489300er_nat(F,A)))
      <=> ! [X3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8642_prod__pos__nat__iff,axiom,
    ! [A: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A)))
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8643_prod__pos__nat__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A)))
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8644_int__prod,axiom,
    ! [F: fun_int_nat,A: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A)) = groups1705073143266064639nt_int(aTP_Lamm_kb(F),A) ).

% int_prod
tff(fact_8645_int__prod,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_nat_int(semiri1314217659103216013at_int,groups708209901874060359at_nat(F,A)) = groups705719431365010083at_int(aTP_Lamm_kc(F),A) ).

% int_prod
tff(fact_8646_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_gq(U)) ).

% atMost_def
tff(fact_8647_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_acu,U)) ).

% atMost_def
tff(fact_8648_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_acv,U)) ).

% atMost_def
tff(fact_8649_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gl,U)) ).

% atMost_def
tff(fact_8650_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_acw,U)) ).

% atMost_def
tff(fact_8651_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_8652_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_8653_not__UNIV__le__Iic,axiom,
    ! [H: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(top_top_set_real),set_ord_atMost_real(H))) ).

% not_UNIV_le_Iic
tff(fact_8654_not__UNIV__le__Iic,axiom,
    ! [H: nat] : ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(top_top_set_nat),aa_nat_set_nat(set_ord_atMost_nat,H))) ).

% not_UNIV_le_Iic
tff(fact_8655_not__UNIV__le__Iic,axiom,
    ! [H: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(top_top_set_int),set_ord_atMost_int(H))) ).

% not_UNIV_le_Iic
tff(fact_8656_not__Iic__le__Icc,axiom,
    ! [H: int,L3: int,H4: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(H)),set_or1266510415728281911st_int(L3,H4))) ).

% not_Iic_le_Icc
tff(fact_8657_not__Iic__le__Icc,axiom,
    ! [H: real,L3: real,H4: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L3,H4))) ).

% not_Iic_le_Icc
tff(fact_8658_finite__nat__iff__bounded__le,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(S),aa_nat_set_nat(set_ord_atMost_nat,K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_8659_frac__ge__0,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X2))) ).

% frac_ge_0
tff(fact_8660_frac__ge__0,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X2))) ).

% frac_ge_0
tff(fact_8661_frac__lt__1,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X2)),one_one_real)) ).

% frac_lt_1
tff(fact_8662_frac__lt__1,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X2)),one_one_rat)) ).

% frac_lt_1
tff(fact_8663_frac__1__eq,axiom,
    ! [X2: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X2) ).

% frac_1_eq
tff(fact_8664_frac__1__eq,axiom,
    ! [X2: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X2),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X2) ).

% frac_1_eq
tff(fact_8665_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8923325533196201883nteger] :
      ~ ! [A3: code_integer,B3: code_integer] : Y3 != produc1086072967326762835nteger(A3,B3) ).

% old.prod.exhaust
tff(fact_8666_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc3658429121746597890et_nat] :
      ~ ! [A3: heap_e7401611519738050253t_unit,B3: set_nat] : Y3 != produc7507926704131184380et_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_8667_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_num_num] :
      ~ ! [A3: num,B3: num] : Y3 != product_Pair_num_num(A3,B3) ).

% old.prod.exhaust
tff(fact_8668_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y3 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) ).

% old.prod.exhaust
tff(fact_8669_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y3 != aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3) ).

% old.prod.exhaust
tff(fact_8670_surj__pair,axiom,
    ! [P2: produc8923325533196201883nteger] :
    ? [X4: code_integer,Y4: code_integer] : P2 = produc1086072967326762835nteger(X4,Y4) ).

% surj_pair
tff(fact_8671_surj__pair,axiom,
    ! [P2: produc3658429121746597890et_nat] :
    ? [X4: heap_e7401611519738050253t_unit,Y4: set_nat] : P2 = produc7507926704131184380et_nat(X4,Y4) ).

% surj_pair
tff(fact_8672_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X4: num,Y4: num] : P2 = product_Pair_num_num(X4,Y4) ).

% surj_pair
tff(fact_8673_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y4: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) ).

% surj_pair
tff(fact_8674_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X4: int,Y4: int] : P2 = aa_int7462565939943104719nt_int(product_Pair_int_int(X4),Y4) ).

% surj_pair
tff(fact_8675_prod__cases,axiom,
    ! [P: fun_Pr3195429412498512255r_bool,P2: produc8923325533196201883nteger] :
      ( ! [A3: code_integer,B3: code_integer] : pp(aa_Pro5843519176478241656r_bool(P,produc1086072967326762835nteger(A3,B3)))
     => pp(aa_Pro5843519176478241656r_bool(P,P2)) ) ).

% prod_cases
tff(fact_8676_prod__cases,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,P2: produc3658429121746597890et_nat] :
      ( ! [A3: heap_e7401611519738050253t_unit,B3: set_nat] : pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(A3,B3)))
     => pp(aa_Pro2991721470636411351t_bool(P,P2)) ) ).

% prod_cases
tff(fact_8677_prod__cases,axiom,
    ! [P: fun_Pr6287530515171585797m_bool,P2: product_prod_num_num] :
      ( ! [A3: num,B3: num] : pp(aa_Pro762926636801165566m_bool(P,product_Pair_num_num(A3,B3)))
     => pp(aa_Pro762926636801165566m_bool(P,P2)) ) ).

% prod_cases
tff(fact_8678_prod__cases,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)))
     => pp(aa_Pro2232559814647311250t_bool(P,P2)) ) ).

% prod_cases
tff(fact_8679_prod__cases,axiom,
    ! [P: fun_Pr803113274115125473t_bool,P2: product_prod_int_int] :
      ( ! [A3: int,B3: int] : pp(aa_Pro4501881432599481050t_bool(P,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3)))
     => pp(aa_Pro4501881432599481050t_bool(P,P2)) ) ).

% prod_cases
tff(fact_8680_Pair__inject,axiom,
    ! [A2: code_integer,B2: code_integer,A4: code_integer,B4: code_integer] :
      ( ( produc1086072967326762835nteger(A2,B2) = produc1086072967326762835nteger(A4,B4) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_8681_Pair__inject,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,A4: heap_e7401611519738050253t_unit,B4: set_nat] :
      ( ( produc7507926704131184380et_nat(A2,B2) = produc7507926704131184380et_nat(A4,B4) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_8682_Pair__inject,axiom,
    ! [A2: num,B2: num,A4: num,B4: num] :
      ( ( product_Pair_num_num(A2,B2) = product_Pair_num_num(A4,B4) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_8683_Pair__inject,axiom,
    ! [A2: nat,B2: nat,A4: nat,B4: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_8684_Pair__inject,axiom,
    ! [A2: int,B2: int,A4: int,B4: int] :
      ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2) = aa_int7462565939943104719nt_int(product_Pair_int_int(A4),B4) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_8685_Iic__subset__Iio__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(A2)),set_ord_lessThan_rat(B2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_8686_Iic__subset__Iio__iff,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(A2)),set_ord_lessThan_num(B2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_8687_Iic__subset__Iio__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A2)),aa_nat_set_nat(set_ord_lessThan_nat,B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_8688_Iic__subset__Iio__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(A2)),set_ord_lessThan_int(B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_8689_Iic__subset__Iio__iff,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(A2)),set_or5984915006950818249n_real(B2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% Iic_subset_Iio_iff
tff(fact_8690_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8691_frac__def,axiom,
    ! [X2: real] : aa_real_real(archim2898591450579166408c_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X2))) ).

% frac_def
tff(fact_8692_frac__def,axiom,
    ! [X2: rat] : aa_rat_rat(archimedean_frac_rat,X2) = aa_rat_rat(minus_minus_rat(X2),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X2))) ).

% frac_def
tff(fact_8693_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_qk(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8694_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8695_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8696_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8697_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_sk(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8698_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_sl(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8699_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : groups6591440286371151544t_real(aTP_Lamm_sm(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8700_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_complex,N: nat,D2: fun_nat_complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(D2),X3),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_complex(C2,I4) = aa_nat_complex(D2,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8701_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_real,N: nat,D2: fun_nat_real] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(D2),X3),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_real(C2,I4) = aa_nat_real(D2,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8702_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8703_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_qk(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8704_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8705_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_pp(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8706_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_8707_sum_Onested__swap_H,axiom,
    ! [A2: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_acz(A2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qs(A2),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_8708_sum_Onested__swap_H,axiom,
    ! [A2: fun_nat_fun_nat_real,N: nat] : groups6591440286371151544t_real(aTP_Lamm_ada(A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qu(A2),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_8709_prod_Onested__swap_H,axiom,
    ! [A2: fun_nat_fun_nat_int,N: nat] : groups705719431365010083at_int(aTP_Lamm_adb(A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_abm(A2),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_8710_prod_Onested__swap_H,axiom,
    ! [A2: fun_nat_fun_nat_nat,N: nat] : groups708209901874060359at_nat(aTP_Lamm_adc(A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_abo(A2),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_8711_frac__eq,axiom,
    ! [X2: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X2) = X2 )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ).

% frac_eq
tff(fact_8712_frac__eq,axiom,
    ! [X2: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X2) = X2 )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),one_one_rat)) ) ) ).

% frac_eq
tff(fact_8713_frac__add,axiom,
    ! [X2: real,Y3: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X2)),aa_real_real(archim2898591450579166408c_real,Y3))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X2)),aa_real_real(archim2898591450579166408c_real,Y3)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X2)),aa_real_real(archim2898591450579166408c_real,Y3))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X2)),aa_real_real(archim2898591450579166408c_real,Y3))),one_one_real) ) ) ) ).

% frac_add
tff(fact_8714_frac__add,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X2)),aa_rat_rat(archimedean_frac_rat,Y3))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X2),Y3)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X2)),aa_rat_rat(archimedean_frac_rat,Y3)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X2)),aa_rat_rat(archimedean_frac_rat,Y3))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X2),Y3)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X2)),aa_rat_rat(archimedean_frac_rat,Y3))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_8715_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_complex,N: nat,K: nat] :
      ( ! [W2: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),W2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_complex(C2,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8716_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_real,N: nat,K: nat] :
      ( ! [W2: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),W2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_real(C2,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8717_polyfun__eq__0,axiom,
    ! [C2: fun_nat_complex,N: nat] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_complex(C2,I4) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8718_polyfun__eq__0,axiom,
    ! [C2: fun_nat_real,N: nat] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_real(C2,I4) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8719_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_qk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8720_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8721_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pp(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8722_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8723_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_8724_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_8725_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_8726_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_8727_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8728_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_qk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8729_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ql(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8730_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_pp(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8731_gbinomial__parallel__sum,axiom,
    ! [A2: rat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_add(A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),N) ).

% gbinomial_parallel_sum
tff(fact_8732_gbinomial__parallel__sum,axiom,
    ! [A2: real,N: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ade,A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),N) ).

% gbinomial_parallel_sum
tff(fact_8733_sum__gp__basic,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8734_sum__gp__basic,axiom,
    ! [X2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),X2)),groups7501900531339628137nteger(power_8256067586552552935nteger(X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8735_sum__gp__basic,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8736_sum__gp__basic,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X2)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8737_sum__gp__basic,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8738_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_complex,K: nat,N: nat] :
      ( ( aa_nat_complex(C2,K) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_adf(C2),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_8739_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_real,K: nat,N: nat] :
      ( ( aa_nat_real(C2,K) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_adg(C2),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_8740_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_adf(C2),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_complex(C2,I4) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8741_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_adg(C2),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_real(C2,I4) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8742_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_complex,A2: complex,N: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z4),A2)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8743_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_Code_integer,A2: code_integer,N: nat] :
      ( ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger )
     => ~ ! [B3: fun_nat_Code_integer] :
            ~ ! [Z4: code_integer] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Z4),A2)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8744_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_rat,A2: rat,N: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z4),A2)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8745_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_int,A2: int,N: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A2)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8746_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_real,A2: real,N: nat] :
      ( ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),A2)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8747_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_complex,N: nat,A2: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z4),A2)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8748_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_Code_integer,N: nat,A2: code_integer] :
    ? [B3: fun_nat_Code_integer] :
    ! [Z4: code_integer] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Z4),A2)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8749_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_rat,N: nat,A2: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z4),A2)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8750_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_int,N: nat,A2: int] :
    ? [B3: fun_nat_int] :
    ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A2)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8751_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_real,N: nat,A2: real] :
    ? [B3: fun_nat_real] :
    ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),A2)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),A2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8752_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),M)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8753_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups7501900531339628137nteger(power_8256067586552552935nteger(X2),set_or1269000886237332187st_nat(M,N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),M)),groups7501900531339628137nteger(power_8256067586552552935nteger(X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8754_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),M)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8755_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),M)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8756_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8757_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_8758_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_rc(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8759_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_rd(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8760_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_re(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8761_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups6591440286371151544t_real(aTP_Lamm_rf(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8762_polynomial__product,axiom,
    ! [M: nat,A2: fun_nat_complex,N: nat,B2: fun_nat_complex,X2: complex] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_complex(A2,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_complex(B2,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(B2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_adl(A2),B2),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8763_polynomial__product,axiom,
    ! [M: nat,A2: fun_nat_Code_integer,N: nat,B2: fun_nat_Code_integer,X2: code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_Code_integer(A2,I2) = zero_z3403309356797280102nteger ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_Code_integer(B2,J2) = zero_z3403309356797280102nteger ) )
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(B2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_fun8469111845059130348nteger(aTP_Lamm_adn(A2),B2),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8764_polynomial__product,axiom,
    ! [M: nat,A2: fun_nat_rat,N: nat,B2: fun_nat_rat,X2: rat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_rat(A2,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_rat(B2,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(B2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_adp(A2),B2),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8765_polynomial__product,axiom,
    ! [M: nat,A2: fun_nat_int,N: nat,B2: fun_nat_int,X2: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_int(A2,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_int(B2,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(B2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_adr(A2),B2),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8766_polynomial__product,axiom,
    ! [M: nat,A2: fun_nat_real,N: nat,B2: fun_nat_real,X2: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_real(A2,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_real(B2,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(B2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_adt(A2),B2),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8767_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups129246275422532515t_real(aTP_Lamm_acj(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8768_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_ack(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8769_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_acl(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8770_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups708209901874060359at_nat(aTP_Lamm_acm(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8771_polyfun__eq__const,axiom,
    ! [C2: fun_nat_complex,N: nat,K: complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_complex(C2,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,N)))
           => ( aa_nat_complex(C2,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8772_polyfun__eq__const,axiom,
    ! [C2: fun_nat_real,N: nat,K: real] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_real(C2,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,N)))
           => ( aa_nat_real(C2,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8773_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun_nat_nat,N: nat,B2: fun_nat_nat,X2: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_nat(A2,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_nat(B2,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_adu(A2),X2)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_adu(B2),X2)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_adw(A2),B2),X2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_8774_floor__add,axiom,
    ! [X2: real,Y3: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X2)),aa_real_real(archim2898591450579166408c_real,Y3))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim6058952711729229775r_real,Y3)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X2)),aa_real_real(archim2898591450579166408c_real,Y3))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X2)),aa_real_int(archim6058952711729229775r_real,Y3))),one_one_int) ) ) ) ).

% floor_add
tff(fact_8775_floor__add,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X2)),aa_rat_rat(archimedean_frac_rat,Y3))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X2),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim3151403230148437115or_rat,Y3)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X2)),aa_rat_rat(archimedean_frac_rat,Y3))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X2),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X2)),aa_rat_int(archim3151403230148437115or_rat,Y3))),one_one_int) ) ) ) ).

% floor_add
tff(fact_8776_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_adx(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ady(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8777_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_adz(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aea(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8778_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aeb(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aec(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8779_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aed(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aee(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8780_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aef(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aeg(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8781_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_assn,H: fun_nat_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aeh(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aei(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8782_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aej(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aeg(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8783_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aek(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aea(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8784_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ael(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aec(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8785_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P2))
       => ( groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aem(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aee(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8786_sum__gp0,axiom,
    ! [X2: complex,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X2 != one_one_complex )
       => ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X2),aa_nat_nat(suc,N)))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),X2)) ) ) ) ).

% sum_gp0
tff(fact_8787_sum__gp0,axiom,
    ! [X2: rat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X2 != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ).

% sum_gp0
tff(fact_8788_sum__gp0,axiom,
    ! [X2: real,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X2 != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ).

% sum_gp0
tff(fact_8789_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2073611262835488442omplex(aTP_Lamm_aen(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8790_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(aTP_Lamm_aeo(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8791_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_aep,M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8792_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A2: complex,X2: complex,Y3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_aeq(M),A2),X2),Y3),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_aer(M),A2),X2),Y3),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8793_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A2: rat,X2: rat,Y3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_aes(M),A2),X2),Y3),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_aet(M),A2),X2),Y3),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8794_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A2: real,X2: real,Y3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_aeu(M),A2),X2),Y3),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_aev(M),A2),X2),Y3),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8795_polyfun__diff__alt,axiom,
    ! [N: nat,A2: fun_nat_complex,X2: complex,Y3: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_aex(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8796_polyfun__diff__alt,axiom,
    ! [N: nat,A2: fun_nat_Code_integer,X2: code_integer,Y3: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),Y3)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aa_fun4629701700493761862nteger(aTP_Lamm_aez(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8797_polyfun__diff__alt,axiom,
    ! [N: nat,A2: fun_nat_rat,X2: rat,Y3: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y3)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_afb(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8798_polyfun__diff__alt,axiom,
    ! [N: nat,A2: fun_nat_int,X2: int,Y3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_afd(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8799_polyfun__diff__alt,axiom,
    ! [N: nat,A2: fun_nat_real,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_aff(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8800_polyfun__extremal__lemma,axiom,
    ! [E: real,C2: fun_nat_complex,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ? [M9: real] :
        ! [Z4: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M9),aa_complex_real(real_V1022390504157884413omplex,Z4)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z4)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8801_polyfun__extremal__lemma,axiom,
    ! [E: real,C2: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ? [M9: real] :
        ! [Z4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M9),aa_real_real(real_V7735802525324610683m_real,Z4)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z4)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8802_polyfun__diff,axiom,
    ! [N: nat,A2: fun_nat_complex,X2: complex,Y3: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_afh(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8803_polyfun__diff,axiom,
    ! [N: nat,A2: fun_nat_Code_integer,X2: code_integer,Y3: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),Y3)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aa_fun4629701700493761862nteger(aTP_Lamm_afj(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8804_polyfun__diff,axiom,
    ! [N: nat,A2: fun_nat_rat,X2: rat,Y3: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y3)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_afl(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8805_polyfun__diff,axiom,
    ! [N: nat,A2: fun_nat_int,X2: int,Y3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_adj(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y3)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_afn(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8806_polyfun__diff,axiom,
    ! [N: nat,A2: fun_nat_real,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(A2),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(A2),Y3),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_afp(N),A2),X2),Y3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8807_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_afq(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8808_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_afr(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8809_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_afs(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8810_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_aft,N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8811_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_afu,N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8812_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_afv(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8813_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_afw(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8814_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_afx(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8815_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_afy,N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8816_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_afz,N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8817_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_8818_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_8819_binomial__n__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ).

% binomial_n_n
tff(fact_8820_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_8821_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8822_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_8823_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8824_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8825_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_8826_choose__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).

% choose_one
tff(fact_8827_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8828_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).

% Suc_times_binomial
tff(fact_8829_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_8830_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_8831_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_8832_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% binomial_le_pow
tff(fact_8833_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).

% zero_less_binomial
tff(fact_8834_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),aa_nat_nat(suc,A2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_8835_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8836_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_8837_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_8838_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aga,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8839_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8840_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agb,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),N))),N) ).

% sum_choose_lower
tff(fact_8841_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agc,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8842_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agc,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8843_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_8844_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_8845_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8846_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8847_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_8848_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K4)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4))) ) ) ).

% binomial_mono
tff(fact_8849_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_8850_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8851_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8852_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agd(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8853_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_age(M),N),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R2) ).

% vandermonde
tff(fact_8854_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_8855_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_8856_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K4)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4))) ) ) ).

% binomial_strict_mono
tff(fact_8857_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8858_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8859_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_8860_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_agf(A2),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_8861_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8862_binomial__ring,axiom,
    ! [A2: complex,B2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)),N) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_agg(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8863_binomial__ring,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),N) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_agh(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8864_binomial__ring,axiom,
    ! [A2: rat,B2: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A2),B2)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_agi(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8865_binomial__ring,axiom,
    ! [A2: int,B2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),N) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_agj(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8866_binomial__ring,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_agf(A2),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8867_binomial__ring,axiom,
    ! [A2: real,B2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),N) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_agk(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8868_binomial__altdef__of__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_agl(K),N),set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_8869_binomial__altdef__of__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_agm(K),N),set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_8870_binomial__altdef__of__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_agn(K),N),set_or4665077453230672383an_nat(zero_zero_nat,K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_8871_pochhammer__binomial__sum,axiom,
    ! [A2: rat,B2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A2),B2)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ago(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8872_pochhammer__binomial__sum,axiom,
    ! [A2: int,B2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),N) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_agp(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8873_pochhammer__binomial__sum,axiom,
    ! [A2: real,B2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),N) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_agq(A2),B2),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8874_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_agr,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_8875_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_8876_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ags,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_8877_upd__rule,axiom,
    ! [I: nat,Xs: list_bool,A2: array_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => hoare_9097894755699397917y_bool(snga_assn_bool(A2,Xs),array_upd_bool(I,X2,A2),aa_boo8717665131543412656l_assn(aa_arr1850246612958046993l_assn(aa_lis1773218934604694151l_assn(aTP_Lamm_agt(I),Xs),A2),X2)) ) ).

% upd_rule
tff(fact_8878_upd__rule,axiom,
    ! [I: nat,Xs: list_nat,A2: array_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => hoare_6807272225193264096ay_nat(snga_assn_nat(A2,Xs),array_upd_nat(I,X2,A2),aa_nat8392338049721622586t_assn(aa_arr4802002299793635008t_assn(aa_lis5007279772662053221t_assn(aTP_Lamm_agu(I),Xs),A2),X2)) ) ).

% upd_rule
tff(fact_8879_upd__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A2: array_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => hoare_3353465787467722821_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A2),Xs),array_upd_VEBT_VEBTi(I,X2,A2),aa_VEB7841561841333234298i_assn(aa_arr6331738488200487899i_assn(aa_lis1967528460011584869i_assn(aTP_Lamm_agv(I),Xs),A2),X2)) ) ).

% upd_rule
tff(fact_8880_gbinomial__code,axiom,
    ! [K: nat,A2: complex] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),K) = one_one_complex ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A2),K) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_agw(A2),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K)) ) ) ) ).

% gbinomial_code
tff(fact_8881_gbinomial__code,axiom,
    ! [K: nat,A2: rat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A2),K) = one_one_rat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A2),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_agx(A2),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ) ) ).

% gbinomial_code
tff(fact_8882_gbinomial__code,axiom,
    ! [K: nat,A2: real] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = one_one_real ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A2),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_agy(A2),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K)) ) ) ) ).

% gbinomial_code
tff(fact_8883_powr__int,axiom,
    ! [X2: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_8884_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_8885_neg__le__iff__le,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2)) ) ).

% neg_le_iff_le
tff(fact_8886_neg__le__iff__le,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ).

% neg_le_iff_le
tff(fact_8887_neg__le__iff__le,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2)) ) ).

% neg_le_iff_le
tff(fact_8888_neg__le__iff__le,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% neg_le_iff_le
tff(fact_8889_neg__less__iff__less,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% neg_less_iff_less
tff(fact_8890_neg__less__iff__less,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% neg_less_iff_less
tff(fact_8891_neg__less__iff__less,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% neg_less_iff_less
tff(fact_8892_neg__less__iff__less,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% neg_less_iff_less
tff(fact_8893_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_8894_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_8895_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_8896_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_8897_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_8898_minus__add__distrib,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) ).

% minus_add_distrib
tff(fact_8899_minus__add__distrib,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) ).

% minus_add_distrib
tff(fact_8900_minus__add__distrib,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_add_distrib
tff(fact_8901_minus__add__distrib,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% minus_add_distrib
tff(fact_8902_minus__add__distrib,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_add_distrib
tff(fact_8903_minus__add__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_8904_minus__add__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_8905_minus__add__cancel,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_8906_minus__add__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_8907_minus__add__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(plus_plus_rat(A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_8908_add__minus__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_8909_add__minus__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_8910_add__minus__cancel,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_8911_add__minus__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_8912_add__minus__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_8913_mult__minus__left,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ).

% mult_minus_left
tff(fact_8914_mult__minus__left,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ).

% mult_minus_left
tff(fact_8915_mult__minus__left,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)) ).

% mult_minus_left
tff(fact_8916_mult__minus__left,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)) ).

% mult_minus_left
tff(fact_8917_mult__minus__left,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A2),B2)) ).

% mult_minus_left
tff(fact_8918_minus__mult__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) ).

% minus_mult_minus
tff(fact_8919_minus__mult__minus,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) ).

% minus_mult_minus
tff(fact_8920_minus__mult__minus,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2) ).

% minus_mult_minus
tff(fact_8921_minus__mult__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2) ).

% minus_mult_minus
tff(fact_8922_minus__mult__minus,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(times_times_rat(A2),B2) ).

% minus_mult_minus
tff(fact_8923_mult__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ).

% mult_minus_right
tff(fact_8924_mult__minus__right,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ).

% mult_minus_right
tff(fact_8925_mult__minus__right,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)) ).

% mult_minus_right
tff(fact_8926_mult__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A2),B2)) ).

% mult_minus_right
tff(fact_8927_mult__minus__right,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A2),B2)) ).

% mult_minus_right
tff(fact_8928_minus__diff__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2) ).

% minus_diff_eq
tff(fact_8929_minus__diff__eq,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2) ).

% minus_diff_eq
tff(fact_8930_minus__diff__eq,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B2),A2) ).

% minus_diff_eq
tff(fact_8931_minus__diff__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),A2) ).

% minus_diff_eq
tff(fact_8932_minus__diff__eq,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A2),B2)) = aa_rat_rat(minus_minus_rat(B2),A2) ).

% minus_diff_eq
tff(fact_8933_div__minus__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ).

% div_minus_minus
tff(fact_8934_div__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) ).

% div_minus_minus
tff(fact_8935_mod__minus__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% mod_minus_minus
tff(fact_8936_mod__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ).

% mod_minus_minus
tff(fact_8937_neg__less__eq__nonneg,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_8938_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_8939_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_8940_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% neg_less_eq_nonneg
tff(fact_8941_less__eq__neg__nonpos,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_8942_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_8943_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_8944_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_8945_neg__le__0__iff__le,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A2)) ) ).

% neg_le_0_iff_le
tff(fact_8946_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_le_0_iff_le
tff(fact_8947_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A2)) ) ).

% neg_le_0_iff_le
tff(fact_8948_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% neg_le_0_iff_le
tff(fact_8949_neg__0__le__iff__le,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_8950_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_8951_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_8952_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_8953_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% neg_less_0_iff_less
tff(fact_8954_neg__less__0__iff__less,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2)) ) ).

% neg_less_0_iff_less
tff(fact_8955_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ).

% neg_less_0_iff_less
tff(fact_8956_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2)) ) ).

% neg_less_0_iff_less
tff(fact_8957_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_8958_neg__0__less__iff__less,axiom,
    ! [A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_8959_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_8960_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_8961_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_8962_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_8963_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = aa_int_int(uminus_uminus_int,A2) )
      <=> ( B2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_8964_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_8965_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_8966_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_8967_int__cases,axiom,
    ! [Z: int] :
      ( ! [N3: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% int_cases
tff(fact_8968_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z: int] :
      ( ! [N3: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N3)))
     => ( ! [N3: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3)))))
       => pp(aa_int_bool(P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_8969_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_8970_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_8971_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_8972_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_8973_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_8974_prod__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_8975_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_8976_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ) ).

% int_cases4
tff(fact_8977_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_8978_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_8979_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N3: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% nonpos_int_cases
tff(fact_8980_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(uminus_uminus_int,B2)) = zero_zero_int ) )
      & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_8981_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) = zero_zero_int ) )
      & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_8982_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_8983_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_8984_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_8985_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) )
       => ~ ! [N3: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ) ).

% int_cases3
tff(fact_8986_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),groups708209901874060359at_nat(aTP_Lamm_ai,set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8987_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_8988_negD,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
     => ? [N3: nat] : X2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% negD
tff(fact_8989_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_8990_verit__less__mono__div__int2,axiom,
    ! [A: int,B: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_8991_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_8992_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8993_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_ai,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_8994_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N3: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% neg_int_cases
tff(fact_8995_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8996_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_8997_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),one_one_int) ) ) ).

% zmod_minus1
tff(fact_8998_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) )
        & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_8999_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) )
        & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_9000_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% square_fact_le_2_fact
tff(fact_9001_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)))
     => ( ( B2 != zero_zero_int )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A2),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)),aa_int_int(uminus_uminus_int,Q3)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q3)),one_one_int))),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),R2))))) ) ) ).

% zminus1_lemma
tff(fact_9002_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_9003_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_9004_int__bit__induct,axiom,
    ! [P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(P,zero_zero_int))
     => ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K2: int] :
              ( pp(aa_int_bool(P,K2))
             => ( ( K2 != zero_zero_int )
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
         => ( ! [K2: int] :
                ( pp(aa_int_bool(P,K2))
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_9005_m1mod2k,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% m1mod2k
tff(fact_9006_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_agz(H),J),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_9007_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2))) ) ).

% sb_dec_lem'
tff(fact_9008_m1mod22k,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),one_one_int) ).

% m1mod22k
tff(fact_9009_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K))))) ) ).

% sb_inc_lem'
tff(fact_9010_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2))) ) ).

% sb_dec_lem
tff(fact_9011_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_9012_one__div__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_9013_minus__one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_9014_real__add__minus__iff,axiom,
    ! [X2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(uminus_uminus_real,A2)) = zero_zero_real )
    <=> ( X2 = A2 ) ) ).

% real_add_minus_iff
tff(fact_9015_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_9016_real__minus__mult__self__le,axiom,
    ! [U: real,X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2))) ).

% real_minus_mult_self_le
tff(fact_9017_minus__real__def,axiom,
    ! [X2: real,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(uminus_uminus_real,Y3)) ).

% minus_real_def
tff(fact_9018_complex__mod__minus__le__complex__mod,axiom,
    ! [X2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X2))),aa_complex_real(real_V1022390504157884413omplex,X2))) ).

% complex_mod_minus_le_complex_mod
tff(fact_9019_real__add__less__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(uminus_uminus_real,X2))) ) ).

% real_add_less_0_iff
tff(fact_9020_real__0__less__add__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X2)),Y3)) ) ).

% real_0_less_add_iff
tff(fact_9021_real__0__le__add__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X2)),Y3)) ) ).

% real_0_le_add_iff
tff(fact_9022_real__add__le__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(uminus_uminus_real,X2))) ) ).

% real_add_le_0_iff
tff(fact_9023_realpow__square__minus__le,axiom,
    ! [U: real,X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_9024_powr__neg__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X2) ) ) ).

% powr_neg_one
tff(fact_9025_Bernoulli__inequality,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)),N))) ) ).

% Bernoulli_inequality
tff(fact_9026_log__minus__eq__powr,axiom,
    ! [B2: real,X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
     => ( ( B2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)),Y3) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_real_real(uminus_uminus_real,Y3)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9027_powr__neg__numeral,axiom,
    ! [X2: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_9028_cos__coeff__def,axiom,
    ! [X6: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X6))
       => ( aa_nat_real(cos_coeff,X6) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X6),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X6)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X6))
       => ( aa_nat_real(cos_coeff,X6) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_9029_sin__coeff__def,axiom,
    ! [X6: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X6))
       => ( aa_nat_real(sin_coeff,X6) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X6))
       => ( aa_nat_real(sin_coeff,X6) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X6),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X6)) ) ) ) ).

% sin_coeff_def
tff(fact_9030_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_9031_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_9032_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_9033_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_9034_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_9035_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_9036_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_9037_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_9038_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_9039_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_9040_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_9041_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_9042_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_9043_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_9044_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_9045_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_9046_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_9047_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_9048_sin__paired,axiom,
    ! [X2: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_aha,X2)),aa_real_real(sin_real,X2))) ).

% sin_paired
tff(fact_9049_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_9050_cos__paired,axiom,
    ! [X2: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ahb,X2)),aa_real_real(cos_real,X2))) ).

% cos_paired
tff(fact_9051_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K = zero_zero_int )
           => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = K ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_9052_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_9053_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_9054_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_9055_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_9056_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_9057_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_9058_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_9059_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_9060_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_9061_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_9062_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_9063_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_9064_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9065_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(6)
tff(fact_9066_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(2)
tff(fact_9067_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_9068_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_9069_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_9070_sin__x__le__x,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X2)),X2)) ) ).

% sin_x_le_x
tff(fact_9071_OR__lower,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X2),Y3))) ) ) ).

% OR_lower
tff(fact_9072_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))) ) ).

% or_greater_eq
tff(fact_9073_sin__le__one,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X2)),one_one_real)) ).

% sin_le_one
tff(fact_9074_cos__le__one,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X2)),one_one_real)) ).

% cos_le_one
tff(fact_9075_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_9076_sin__x__ge__neg__x,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X2)),aa_real_real(sin_real,X2))) ) ).

% sin_x_ge_neg_x
tff(fact_9077_sin__ge__minus__one,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X2))) ).

% sin_ge_minus_one
tff(fact_9078_cos__ge__minus__one,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X2))) ).

% cos_ge_minus_one
tff(fact_9079_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_9080_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_9081_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9082_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_9083_sin__gt__zero__02,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X2))) ) ) ).

% sin_gt_zero_02
tff(fact_9084_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_9085_cos__is__zero,axiom,
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X4) = zero_zero_real )
      & ! [Y5: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y5),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y5) = zero_zero_real ) )
         => ( Y5 = X4 ) ) ) ).

% cos_is_zero
tff(fact_9086_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_9087_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(M,aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_9088_OR__upper,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X2),Y3)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_9089_cos__double__less__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_9090_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T7))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),X2))
            & ( aa_real_real(cos_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahc,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9091_sin__pi__minus,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),X2)) = aa_real_real(sin_real,X2) ).

% sin_pi_minus
tff(fact_9092_or__nat__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_nat_numerals(2)
tff(fact_9093_or__nat__numerals_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2)) ).

% or_nat_numerals(4)
tff(fact_9094_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_9095_cos__periodic__pi,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X2)) ).

% cos_periodic_pi
tff(fact_9096_cos__periodic__pi2,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X2)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X2)) ).

% cos_periodic_pi2
tff(fact_9097_sin__periodic__pi,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X2)) ).

% sin_periodic_pi
tff(fact_9098_sin__periodic__pi2,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X2)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X2)) ).

% sin_periodic_pi2
tff(fact_9099_cos__pi__minus,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),X2)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X2)) ).

% cos_pi_minus
tff(fact_9100_cos__minus__pi,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X2)) ).

% cos_minus_pi
tff(fact_9101_sin__minus__pi,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X2)) ).

% sin_minus_pi
tff(fact_9102_or__nat__numerals_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2)) ).

% or_nat_numerals(3)
tff(fact_9103_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% or_nat_numerals(1)
tff(fact_9104_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_9105_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_9106_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_9107_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_9108_cos__periodic,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X2) ).

% cos_periodic
tff(fact_9109_sin__periodic,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X2) ).

% sin_periodic
tff(fact_9110_cos__2pi__minus,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X2)) = aa_real_real(cos_real,X2) ).

% cos_2pi_minus
tff(fact_9111_cos__npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi
tff(fact_9112_cos__npi2,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi2
tff(fact_9113_sin__2npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_9114_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_9115_sin__2pi__minus,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X2)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X2)) ).

% sin_2pi_minus
tff(fact_9116_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_9117_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).

% cos_int_2pin
tff(fact_9118_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_9119_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_9120_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = one_one_real ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% cos_npi_int
tff(fact_9121_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_9122_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_9123_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_9124_cos__monotone__0__pi__le,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X2)),aa_real_real(cos_real,Y3))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_9125_cos__mono__le__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X2)),aa_real_real(cos_real,Y3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_9126_cos__inj__pi,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
           => ( ( aa_real_real(cos_real,X2) = aa_real_real(cos_real,Y3) )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_9127_sin__ge__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X2))) ) ) ).

% sin_ge_zero
tff(fact_9128_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_9129_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) ).

% pi_ge_two
tff(fact_9130_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_9131_cos__monotone__0__pi,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X2)),aa_real_real(cos_real,Y3))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_9132_cos__mono__less__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X2)),aa_real_real(cos_real,Y3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_9133_cos__monotone__minus__pi__0_H,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y3)),aa_real_real(cos_real,X2))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_9134_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_9135_pi__half__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_9136_pi__half__le__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_le_two
tff(fact_9137_cos__monotone__minus__pi__0,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y3)),aa_real_real(cos_real,X2))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_9138_cos__total,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),pi))
            & ( aa_real_real(cos_real,X4) = Y3 )
            & ! [Y5: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y5),pi))
                  & ( aa_real_real(cos_real,Y5) = Y3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_9139_sincos__principal__value,axiom,
    ! [X2: real] :
    ? [Y4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),pi))
      & ( aa_real_real(sin_real,Y4) = aa_real_real(sin_real,X2) )
      & ( aa_real_real(cos_real,Y4) = aa_real_real(cos_real,X2) ) ) ).

% sincos_principal_value
tff(fact_9140_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_9141_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_ge_zero
tff(fact_9142_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_9143_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_9144_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_9145_cos__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X2))) ) ) ).

% cos_gt_zero
tff(fact_9146_sin__gt__zero2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X2))) ) ) ).

% sin_gt_zero2
tff(fact_9147_sin__lt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X2)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_9148_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_9149_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_9150_sin__monotone__2pi__le,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y3)),aa_real_real(sin_real,X2))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_9151_sin__mono__le__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X2)),aa_real_real(sin_real,Y3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_9152_sin__inj__pi,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( ( aa_real_real(sin_real,X2) = aa_real_real(sin_real,Y3) )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_9153_cos__one__2pi__int,axiom,
    ! [X2: real] :
      ( ( aa_real_real(cos_real,X2) = one_one_real )
    <=> ? [X3: int] : X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_9154_sin__le__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X2)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_9155_cos__gt__zero__pi,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X2))) ) ) ).

% cos_gt_zero_pi
tff(fact_9156_sin__less__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X2)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_9157_cos__ge__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X2))) ) ) ).

% cos_ge_zero
tff(fact_9158_sin__monotone__2pi,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y3)),aa_real_real(sin_real,X2))) ) ) ) ).

% sin_monotone_2pi
tff(fact_9159_sin__mono__less__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X2)),aa_real_real(sin_real,Y3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_9160_sin__total,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(sin_real,X4) = Y3 )
            & ! [Y5: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
                  & ( aa_real_real(sin_real,Y5) = Y3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_9161_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_9162_cos__one__2pi,axiom,
    ! [X2: real] :
      ( ( aa_real_real(cos_real,X2) = one_one_real )
    <=> ( ? [X3: nat] : X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X3: nat] : X2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_9163_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_9164_sincos__total__pi,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),pi))
            & ( X2 = aa_real_real(cos_real,T7) )
            & ( Y3 = aa_real_real(sin_real,T7) ) ) ) ) ).

% sincos_total_pi
tff(fact_9165_sin__expansion__lemma,axiom,
    ! [X2: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_9166_cos__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( aa_real_real(cos_real,X2) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_9167_sin__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( aa_real_real(sin_real,X2) = zero_zero_real )
    <=> ? [I4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_9168_cos__zero__lemma,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( ( aa_real_real(cos_real,X2) = zero_zero_real )
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_9169_sin__zero__lemma,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( ( aa_real_real(sin_real,X2) = zero_zero_real )
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_9170_cos__zero__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(cos_real,X2) = zero_zero_real )
    <=> ( ? [N7: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))
            & ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N7: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))
            & ( X2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_9171_sin__zero__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(sin_real,X2) = zero_zero_real )
    <=> ( ? [N7: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))
            & ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N7: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))
            & ( X2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_9172_cos__expansion__lemma,axiom,
    ! [X2: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_9173_sincos__total__pi__half,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T7: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
              & ( X2 = aa_real_real(cos_real,T7) )
              & ( Y3 = aa_real_real(sin_real,T7) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9174_sincos__total__2pi__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T7: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
          & ( X2 = aa_real_real(cos_real,T7) )
          & ( Y3 = aa_real_real(sin_real,T7) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9175_sincos__total__2pi,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X2 = aa_real_real(cos_real,T7) )
               => ( Y3 != aa_real_real(sin_real,T7) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_9176_Maclaurin__sin__expansion,axiom,
    ! [X2: real,N: nat] :
    ? [T7: real] : aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahd,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ).

% Maclaurin_sin_expansion
tff(fact_9177_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ? [T7: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T7))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),X2))
          & ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahd,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9178_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T7))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),X2))
            & ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahd,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9179_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T7))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),zero_zero_real))
            & ( aa_real_real(cos_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahc,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9180_Maclaurin__cos__expansion,axiom,
    ! [X2: real,N: nat] :
    ? [T7: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
      & ( aa_real_real(cos_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahc,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9181_Maclaurin__sin__expansion2,axiom,
    ! [X2: real,N: nat] :
    ? [T7: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
      & ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahd,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9182_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ahe) ).

% pi_series
tff(fact_9183_tan__periodic__pi,axiom,
    ! [X2: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),pi)) = aa_real_real(tan_real,X2) ).

% tan_periodic_pi
tff(fact_9184_tan__periodic__n,axiom,
    ! [X2: real,N: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,N)),pi))) = aa_real_real(tan_real,X2) ).

% tan_periodic_n
tff(fact_9185_tan__periodic__nat,axiom,
    ! [X2: real,N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi))) = aa_real_real(tan_real,X2) ).

% tan_periodic_nat
tff(fact_9186_tan__periodic__int,axiom,
    ! [X2: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,X2) ).

% tan_periodic_int
tff(fact_9187_tan__periodic,axiom,
    ! [X2: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X2) ).

% tan_periodic
tff(fact_9188_square__powr__half,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X2) ).

% square_powr_half
tff(fact_9189_abs__sin__x__le__abs__x,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X2))),aa_real_real(abs_abs_real,X2))) ).

% abs_sin_x_le_abs_x
tff(fact_9190_lemma__interval__lt,axiom,
    ! [A2: real,X2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B2))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [Y5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y5))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Y5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y5),B2)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_9191_abs__cos__le__one,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X2))),one_one_real)) ).

% abs_cos_le_one
tff(fact_9192_abs__sin__le__one,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X2))),one_one_real)) ).

% abs_sin_le_one
tff(fact_9193_sin__bound__lemma,axiom,
    ! [X2: real,Y3: real,U: real,V: real] :
      ( ( X2 = Y3 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),U)),Y3))),V)) ) ) ).

% sin_bound_lemma
tff(fact_9194_tan__bound__pi2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X2))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_9195_tan__total__pi4,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z2) = X2 ) ) ) ).

% tan_total_pi4
tff(fact_9196_lemma__interval,axiom,
    ! [A2: real,X2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B2))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [Y5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y5))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Y5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y5),B2)) ) ) ) ) ) ).

% lemma_interval
tff(fact_9197_sin__zero__abs__cos__one,axiom,
    ! [X2: real] :
      ( ( aa_real_real(sin_real,X2) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X2)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_9198_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_9199_sin__cos__le1,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X2)),aa_real_real(sin_real,Y3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X2)),aa_real_real(cos_real,Y3))))),one_one_real)) ).

% sin_cos_le1
tff(fact_9200_lemma__tan__total,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(tan_real,X4))) ) ) ).

% lemma_tan_total
tff(fact_9201_tan__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X2))) ) ) ).

% tan_gt_zero
tff(fact_9202_tan__total,axiom,
    ! [Y3: real] :
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y3 )
      & ! [Y5: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y5) = Y3 ) )
         => ( Y5 = X4 ) ) ) ).

% tan_total
tff(fact_9203_tan__monotone,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y3)),aa_real_real(tan_real,X2))) ) ) ) ).

% tan_monotone
tff(fact_9204_tan__monotone_H,axiom,
    ! [Y3: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y3)),aa_real_real(tan_real,X2))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_9205_tan__mono__lt__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X2)),aa_real_real(tan_real,Y3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_9206_lemma__tan__total1,axiom,
    ! [Y3: real] :
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y3 ) ) ).

% lemma_tan_total1
tff(fact_9207_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_9208_tan__inverse,axiom,
    ! [Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y3)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y3)) ).

% tan_inverse
tff(fact_9209_tan__total__pos,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X4) = Y3 ) ) ) ).

% tan_total_pos
tff(fact_9210_tan__pos__pi2__le,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X2))) ) ) ).

% tan_pos_pi2_le
tff(fact_9211_tan__less__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X2)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_9212_tan__mono__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X2)),aa_real_real(tan_real,Y3))) ) ) ) ).

% tan_mono_le
tff(fact_9213_tan__mono__le__eq,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X2)),aa_real_real(tan_real,Y3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9214_monoseq__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_ahf,X2)) ) ).

% monoseq_arctan_series
tff(fact_9215_arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( aa_real_real(arctan,X2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ahg,X2)) ) ) ).

% arctan_series
tff(fact_9216_ln__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ahh,X2)) ) ) ) ).

% ln_series
tff(fact_9217_summable__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ahg,X2)) ) ).

% summable_arctan_series
tff(fact_9218_zdvd1__eq,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X2) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_9219_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_9220_arctan__le__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_9221_zero__le__arctan__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% zero_le_arctan_iff
tff(fact_9222_ln__le__cancel__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),aa_real_real(ln_ln_real,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_9223_ln__eq__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( aa_real_real(ln_ln_real,X2) = zero_zero_real )
      <=> ( X2 = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_9224_ln__gt__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ).

% ln_gt_zero_iff
tff(fact_9225_ln__less__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_9226_ln__ge__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ).

% ln_ge_zero_iff
tff(fact_9227_ln__le__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_9228_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ahi(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_9229_arctan__monotone_H,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X2)),aa_real_real(arctan,Y3))) ) ).

% arctan_monotone'
tff(fact_9230_arctan__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X2)),aa_real_real(arctan,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% arctan_le_iff
tff(fact_9231_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_9232_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S))
    <=> ! [M7: int] :
        ? [N7: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M7),aa_int_int(abs_abs_int,N7)))
          & pp(aa_set_int_bool(member_int(N7),S)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_9233_ln__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),X2)) ) ).

% ln_bound
tff(fact_9234_ln__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X2))) ) ).

% ln_gt_zero
tff(fact_9235_ln__less__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X2)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_9236_ln__gt__zero__imp__gt__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_9237_ln__ge__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X2))) ) ).

% ln_ge_zero
tff(fact_9238_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_9239_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9240_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D2)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_9241_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N9: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N3))),aa_nat_real(G,N3))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ahi(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_9242_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ahi(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_ahi(F)))) ) ).

% summable_rabs
tff(fact_9243_ln__ge__zero__imp__ge__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_9244_ln__add__one__self__le__self,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2))),X2)) ) ).

% ln_add_one_self_le_self
tff(fact_9245_ln__mult,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X2)),aa_real_real(ln_ln_real,Y3)) ) ) ) ).

% ln_mult
tff(fact_9246_ln__eq__minus__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( aa_real_real(ln_ln_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real) )
       => ( X2 = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_9247_ln__div,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X2)),aa_real_real(ln_ln_real,Y3)) ) ) ) ).

% ln_div
tff(fact_9248_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_9249_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_9250_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9251_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_9252_ln__le__minus__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_9253_ln__diff__le,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X2)),aa_real_real(ln_ln_real,Y3))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)),Y3))) ) ) ).

% ln_diff_le
tff(fact_9254_ln__add__one__self__le__self2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2))),X2)) ) ).

% ln_add_one_self_le_self2
tff(fact_9255_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).

% even_add_abs_iff
tff(fact_9256_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).

% even_abs_add_iff
tff(fact_9257_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_9258_ln__one__minus__pos__upper__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2))),aa_real_real(uminus_uminus_real,X2))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_9259_ln__powr__bound,axiom,
    ! [X2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A2)),A2))) ) ) ).

% ln_powr_bound
tff(fact_9260_ln__powr__bound2,axiom,
    ! [X2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X2)),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),A2)),X2))) ) ) ).

% ln_powr_bound2
tff(fact_9261_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
         => ( ( B2 != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_real_real(ln_ln_real,A2))),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_9262_monoseq__realpow,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X2)) ) ) ).

% monoseq_realpow
tff(fact_9263_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real))
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_acy(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_9264_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_9265_decr__lemma,axiom,
    ! [D2: int,X2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Z))),one_one_int)),D2))),Z)) ) ).

% decr_lemma
tff(fact_9266_incr__lemma,axiom,
    ! [D2: int,Z: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Z))),one_one_int)),D2)))) ) ).

% incr_lemma
tff(fact_9267_arctan__ubound,axiom,
    ! [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_9268_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_9269_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_9270_arctan__lbound,axiom,
    ! [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y3))) ).

% arctan_lbound
tff(fact_9271_arctan__bounded,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y3)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_9272_arctan,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y3)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y3)) = Y3 ) ) ).

% arctan
tff(fact_9273_arctan__tan,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X2)) = X2 ) ) ) ).

% arctan_tan
tff(fact_9274_arctan__unique,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X2) = Y3 )
         => ( aa_real_real(arctan,Y3) = X2 ) ) ) ) ).

% arctan_unique
tff(fact_9275_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_9276_ln__one__plus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_9277_arctan__add,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X2)),aa_real_real(arctan,Y3)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3)))) ) ) ) ).

% arctan_add
tff(fact_9278_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2))),X2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_9279_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_9280_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_9281_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_9282_ln__one__minus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_9283_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2))),X2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_9284_arctan__double,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X2)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_9285_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2))),X2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_9286_tanh__ln__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_9287_sin__tan,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X2)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_9288_real__sqrt__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X2)),aa_real_real(sqrt,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% real_sqrt_le_iff
tff(fact_9289_real__sqrt__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(sqrt,X2) = one_one_real )
    <=> ( X2 = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_9290_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_9291_tanh__real__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X2)),aa_real_real(tanh_real,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% tanh_real_le_iff
tff(fact_9292_real__sqrt__ge__0__iff,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3)) ) ).

% real_sqrt_ge_0_iff
tff(fact_9293_real__sqrt__le__0__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_9294_real__sqrt__gt__1__iff,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y3)) ) ).

% real_sqrt_gt_1_iff
tff(fact_9295_real__sqrt__lt__1__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X2)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_9296_real__sqrt__le__1__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X2)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ).

% real_sqrt_le_1_iff
tff(fact_9297_real__sqrt__ge__1__iff,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y3)) ) ).

% real_sqrt_ge_1_iff
tff(fact_9298_tanh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_9299_tanh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% tanh_real_nonneg_iff
tff(fact_9300_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_9301_artanh__minus__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X2)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X2)) ) ) ).

% artanh_minus_real
tff(fact_9302_real__sqrt__abs,axiom,
    ! [X2: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X2) ).

% real_sqrt_abs
tff(fact_9303_real__sqrt__pow2__iff,axiom,
    ! [X2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X2 )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% real_sqrt_pow2_iff
tff(fact_9304_real__sqrt__pow2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X2 ) ) ).

% real_sqrt_pow2
tff(fact_9305_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X2: real,Y3: real,Xa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_9306_real__sqrt__le__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X2)),aa_real_real(sqrt,Y3))) ) ).

% real_sqrt_le_mono
tff(fact_9307_real__sqrt__ge__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X2))) ) ).

% real_sqrt_ge_zero
tff(fact_9308_real__sqrt__eq__zero__cancel,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( ( aa_real_real(sqrt,X2) = zero_zero_real )
       => ( X2 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_9309_real__sqrt__ge__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X2))) ) ).

% real_sqrt_ge_one
tff(fact_9310_tanh__real__lt__1,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X2)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_9311_real__div__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_real_real(sqrt,X2)) = aa_real_real(sqrt,X2) ) ) ).

% real_div_sqrt
tff(fact_9312_sqrt__add__le__add__sqrt,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X2)),aa_real_real(sqrt,Y3)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9313_le__real__sqrt__sumsq,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y3),Y3))))) ).

% le_real_sqrt_sumsq
tff(fact_9314_tanh__real__gt__neg1,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X2))) ).

% tanh_real_gt_neg1
tff(fact_9315_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_9316_real__less__rsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(sqrt,Y3))) ) ).

% real_less_rsqrt
tff(fact_9317_real__le__rsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(sqrt,Y3))) ) ).

% real_le_rsqrt
tff(fact_9318_sqrt__le__D,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X2)),Y3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% sqrt_le_D
tff(fact_9319_real__sqrt__unique,axiom,
    ! [Y3: real,X2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( aa_real_real(sqrt,X2) = Y3 ) ) ) ).

% real_sqrt_unique
tff(fact_9320_real__le__lsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X2)),Y3)) ) ) ) ).

% real_le_lsqrt
tff(fact_9321_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_9322_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y3 )
     => ( X2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_9323_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X2 )
     => ( Y3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_9324_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_9325_real__sqrt__sum__squares__ge2,axiom,
    ! [Y3: real,X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9326_real__sqrt__sum__squares__ge1,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9327_sqrt__ge__absD,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),aa_real_real(sqrt,Y3)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y3)) ) ).

% sqrt_ge_absD
tff(fact_9328_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_9329_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_9330_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_9331_real__less__lsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X2)),Y3)) ) ) ) ).

% real_less_lsqrt
tff(fact_9332_sqrt__sum__squares__le__sum,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9333_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_9334_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,X2)),aa_real_real(abs_abs_real,Y3)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9335_real__sqrt__ge__abs2,axiom,
    ! [Y3: real,X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs2
tff(fact_9336_real__sqrt__ge__abs1,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs1
tff(fact_9337_ln__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_9338_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_9339_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_9340_real__sqrt__power__even,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X2)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_9341_arsinh__real__aux,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_9342_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X2: real,Y3: real,Xa: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_9343_arith__geo__mean__sqrt,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y3))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9344_powr__half__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X2) ) ) ).

% powr_half_sqrt
tff(fact_9345_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_9346_cos__x__y__le__one,axiom,
    ! [X2: real,Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real)) ).

% cos_x_y_le_one
tff(fact_9347_real__sqrt__sum__squares__less,axiom,
    ! [X2: real,U: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_9348_arcosh__real__def,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => ( aa_real_real(arcosh_real,X2) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_9349_cos__arctan,axiom,
    ! [X2: real] : aa_real_real(cos_real,aa_real_real(arctan,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_9350_sin__arctan,axiom,
    ! [X2: real] : aa_real_real(sin_real,aa_real_real(arctan,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_9351_sqrt__sum__squares__half__less,axiom,
    ! [X2: real,U: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_9352_sin__cos__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X2)))
     => ( aa_real_real(sin_real,X2) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9353_arctan__half,axiom,
    ! [X2: real] : aa_real_real(arctan,X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_9354_cos__tan,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(cos_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9355_arsinh__real__def,axiom,
    ! [X2: real] : aa_real_real(arsinh_real,X2) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_9356_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T7)),aa_real_real(sin_real,T7)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9357_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_ahj(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_9358_norm__cos__sin,axiom,
    ! [T: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T))) = one_one_real ).

% norm_cos_sin
tff(fact_9359_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_real_complex(complex2(A2),B2)),aa_real_complex(complex2(C2),D2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2)) ).

% complex_diff
tff(fact_9360_complex__of__real__add__Complex,axiom,
    ! [R2: real,X2: real,Y3: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(X2),Y3)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),X2)),Y3) ).

% complex_of_real_add_Complex
tff(fact_9361_Complex__add__complex__of__real,axiom,
    ! [X2: real,Y3: real,R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(complex2(X2),Y3)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),R2)),Y3) ).

% Complex_add_complex_of_real
tff(fact_9362_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(complex2(A2),B2)),aa_real_complex(complex2(C2),D2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) ).

% complex_add
tff(fact_9363_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(A2),B2)),aa_real_complex(complex2(C2),D2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),D2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ).

% complex_mult
tff(fact_9364_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_complex(complex2(A2),B2) = one_one_complex )
    <=> ( ( A2 = one_one_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9365_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9366_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_complex(complex2(A2),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A2 = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9367_complex__norm,axiom,
    ! [X2: real,Y3: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(X2),Y3)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9368_Maclaurin__exp__lt,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T7)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
            & ( aa_real_real(exp_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahk,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9369_cos__arcsin,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X2)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_9370_sin__arccos__abs,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y3)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9371_exp__le__cancel__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X2)),aa_real_real(exp_real,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% exp_le_cancel_iff
tff(fact_9372_exp__eq__one__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(exp_real,X2) = one_one_real )
    <=> ( X2 = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_9373_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9374_one__less__exp__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% one_less_exp_iff
tff(fact_9375_exp__less__one__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X2)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_9376_one__le__exp__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% one_le_exp_iff
tff(fact_9377_exp__le__one__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X2)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_9378_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_9379_cos__arccos,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y3)) = Y3 ) ) ) ).

% cos_arccos
tff(fact_9380_sin__arcsin,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y3)) = Y3 ) ) ) ).

% sin_arcsin
tff(fact_9381_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_9382_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_9383_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_9384_not__exp__le__zero,axiom,
    ! [X2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X2)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_9385_exp__ge__zero,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X2))) ).

% exp_ge_zero
tff(fact_9386_exp__gt__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X2))) ) ).

% exp_gt_one
tff(fact_9387_exp__ge__add__one__self,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)),aa_real_real(exp_real,X2))) ).

% exp_ge_add_one_self
tff(fact_9388_log__ln,axiom,
    ! [X2: real] : aa_real_real(ln_ln_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),X2) ).

% log_ln
tff(fact_9389_arccos__le__arccos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y3)),aa_real_real(arccos,X2))) ) ) ) ).

% arccos_le_arccos
tff(fact_9390_arccos__eq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real)) )
     => ( ( aa_real_real(arccos,X2) = aa_real_real(arccos,Y3) )
      <=> ( X2 = Y3 ) ) ) ).

% arccos_eq_iff
tff(fact_9391_arccos__le__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,X2)),aa_real_real(arccos,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2)) ) ) ) ).

% arccos_le_mono
tff(fact_9392_arcsin__le__arcsin,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X2)),aa_real_real(arcsin,Y3))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9393_arcsin__minus,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X2)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X2)) ) ) ) ).

% arcsin_minus
tff(fact_9394_arcsin__eq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
       => ( ( aa_real_real(arcsin,X2) = aa_real_real(arcsin,Y3) )
        <=> ( X2 = Y3 ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9395_arcsin__le__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X2)),aa_real_real(arcsin,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ) ).

% arcsin_le_mono
tff(fact_9396_exp__ge__add__one__self__aux,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)),aa_real_real(exp_real,X2))) ) ).

% exp_ge_add_one_self_aux
tff(fact_9397_lemma__exp__total,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y3))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y3),one_one_real)))
          & ( aa_real_real(exp_real,X4) = Y3 ) ) ) ).

% lemma_exp_total
tff(fact_9398_ln__ge__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(ln_ln_real,X2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y3)),X2)) ) ) ).

% ln_ge_iff
tff(fact_9399_ln__x__over__x__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y3)),Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X2)),X2))) ) ) ).

% ln_x_over_x_mono
tff(fact_9400_arccos__lbound,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y3))) ) ) ).

% arccos_lbound
tff(fact_9401_arccos__less__arccos,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y3)),aa_real_real(arccos,X2))) ) ) ) ).

% arccos_less_arccos
tff(fact_9402_arccos__less__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X2)),aa_real_real(arccos,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ) ).

% arccos_less_mono
tff(fact_9403_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_9404_arccos__ubound,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y3)),pi)) ) ) ).

% arccos_ubound
tff(fact_9405_arccos__cos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X2)) = X2 ) ) ) ).

% arccos_cos
tff(fact_9406_arcsin__less__arcsin,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X2)),aa_real_real(arcsin,Y3))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9407_arcsin__less__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X2)),aa_real_real(arcsin,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ) ).

% arcsin_less_mono
tff(fact_9408_cos__arccos__abs,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y3)),one_one_real))
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y3)) = Y3 ) ) ).

% cos_arccos_abs
tff(fact_9409_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi))
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_9410_exp__half__le2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% exp_half_le2
tff(fact_9411_arccos__lt__bounded,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y3)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_9412_arccos__bounded,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y3)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_9413_sin__arccos__nonzero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X2)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9414_arccos__cos2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X2))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X2)) = aa_real_real(uminus_uminus_real,X2) ) ) ) ).

% arccos_cos2
tff(fact_9415_arccos__minus,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X2)) ) ) ) ).

% arccos_minus
tff(fact_9416_cos__arcsin__nonzero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X2)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9417_arccos,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y3)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y3)) = Y3 ) ) ) ) ).

% arccos
tff(fact_9418_arccos__minus__abs,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X2)) ) ) ).

% arccos_minus_abs
tff(fact_9419_exp__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_9420_real__exp__bound__lemma,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X2)))) ) ) ).

% real_exp_bound_lemma
tff(fact_9421_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X2))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_9422_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X2)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_9423_arccos__le__pi2,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_9424_arcsin__lt__bounded,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9425_arcsin__bounded,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_bounded
tff(fact_9426_arcsin__ubound,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arcsin_ubound
tff(fact_9427_arcsin__lbound,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3))) ) ) ).

% arcsin_lbound
tff(fact_9428_Maclaurin__exp__le,axiom,
    ! [X2: real,N: nat] :
    ? [T7: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
      & ( aa_real_real(exp_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahk,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_9429_exp__lower__Taylor__quadratic,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,X2))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9430_arcsin__sin,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X2)) = X2 ) ) ) ).

% arcsin_sin
tff(fact_9431_log__base__10__eq2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X2)) ) ) ).

% log_base_10_eq2
tff(fact_9432_tanh__real__altdef,axiom,
    ! [X2: real] : aa_real_real(tanh_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2)))) ).

% tanh_real_altdef
tff(fact_9433_log__base__10__eq1,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X2)) ) ) ).

% log_base_10_eq1
tff(fact_9434_le__arcsin__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(arcsin,X2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y3)),X2)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9435_arcsin__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X2)),Y3))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(sin_real,Y3))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9436_arcsin__pi,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y3)),pi))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y3)) = Y3 ) ) ) ) ).

% arcsin_pi
tff(fact_9437_arcsin,axiom,
    ! [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y3)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y3)) = Y3 ) ) ) ) ).

% arcsin
tff(fact_9438_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9439_sin__arccos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X2)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_9440_Maclaurin__sin__bound,axiom,
    ! [X2: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,X2)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ahd,X2),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X2)),N)))) ).

% Maclaurin_sin_bound
tff(fact_9441_cot__less__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X2)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_9442_cot__periodic,axiom,
    ! [X2: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X2) ).

% cot_periodic
tff(fact_9443_inverse__powr,axiom,
    ! [Y3: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y3)),A2) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y3),A2)) ) ) ).

% inverse_powr
tff(fact_9444_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D4: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E2))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [N3: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono_1
tff(fact_9445_sqrt__divide__self__eq,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X2)),X2) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9446_log__inverse,axiom,
    ! [A2: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
     => ( ( A2 != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(inverse_inverse_real,X2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),X2)) ) ) ) ) ).

% log_inverse
tff(fact_9447_exp__plus__inverse__exp,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X2)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X2))))) ).

% exp_plus_inverse_exp
tff(fact_9448_plus__inverse__ge__2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(inverse_inverse_real,X2)))) ) ).

% plus_inverse_ge_2
tff(fact_9449_real__inv__sqrt__pow2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X2) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9450_tan__cot,axiom,
    ! [X2: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X2)) ).

% tan_cot
tff(fact_9451_real__le__x__sinh,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X2)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_9452_real__le__abs__sinh,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X2)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_9453_powr__real__of__int,axiom,
    ! [X2: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_int_nat(nat2,N)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_9454_cot__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X2))) ) ) ).

% cot_gt_zero
tff(fact_9455_tan__cot_H,axiom,
    ! [X2: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X2)) = aa_real_real(cot_real,X2) ).

% tan_cot'
tff(fact_9456_sinh__real__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X2)),aa_real_real(sinh_real,Y3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ).

% sinh_real_le_iff
tff(fact_9457_sinh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% sinh_real_nonneg_iff
tff(fact_9458_sinh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_9459_sinh__le__cosh__real,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X2)),aa_real_real(cosh_real,X2))) ).

% sinh_le_cosh_real
tff(fact_9460_cosh__real__nonneg,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X2))) ).

% cosh_real_nonneg
tff(fact_9461_cosh__real__nonneg__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X2)),aa_real_real(cosh_real,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9462_cosh__real__nonpos__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X2)),aa_real_real(cosh_real,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X2)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9463_cosh__real__ge__1,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X2))) ).

% cosh_real_ge_1
tff(fact_9464_cosh__real__nonpos__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X2)),aa_real_real(cosh_real,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9465_cosh__real__nonneg__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X2)),aa_real_real(cosh_real,Y3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9466_cosh__real__strict__mono,axiom,
    ! [X2: real,Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X2)),aa_real_real(cosh_real,Y3))) ) ) ).

% cosh_real_strict_mono
tff(fact_9467_arcosh__cosh__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X2)) = X2 ) ) ).

% arcosh_cosh_real
tff(fact_9468_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A2),B2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9469_cosh__ln__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(inverse_inverse_real,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9470_sinh__ln__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(inverse_inverse_real,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9471_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9472_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_9473_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_9474_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_9475_i__squared,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_9476_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_9477_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_9478_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9479_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9480_power2__i,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9481_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9482_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_9483_semiring__norm_I28_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).

% semiring_norm(28)
tff(fact_9484_semiring__norm_I27_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(bitM,N)) ).

% semiring_norm(27)
tff(fact_9485_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_9486_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_9487_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_9488_Complex__eq__i,axiom,
    ! [X2: real,Y3: real] :
      ( ( aa_real_complex(complex2(X2),Y3) = imaginary_unit )
    <=> ( ( X2 = zero_zero_real )
        & ( Y3 = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9489_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9490_Complex__eq,axiom,
    ! [A2: real,B2: real] : aa_real_complex(complex2(A2),B2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B2))) ).

% Complex_eq
tff(fact_9491_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A3: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_9492_cmod__unit__one,axiom,
    ! [A2: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A2))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A2))))) = one_one_real ).

% cmod_unit_one
tff(fact_9493_cmod__complex__polar,axiom,
    ! [R2: real,A2: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A2))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A2)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9494_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9495_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9496_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9497_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_9498_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_9499_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9500_of__real__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X2)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X2)) ) ) ).

% of_real_sqrt
tff(fact_9501_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ).

% Arg_bounded
tff(fact_9502_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9503_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,N)))) ).

% or_minus_numerals(5)
tff(fact_9504_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,N)))) ).

% or_minus_numerals(1)
tff(fact_9505_norm__cis,axiom,
    ! [A2: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A2)) = one_one_real ).

% norm_cis
tff(fact_9506_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9507_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_9508_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_9509_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_9510_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,N)))) ).

% or_minus_numerals(3)
tff(fact_9511_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,N)))) ).

% or_minus_numerals(7)
tff(fact_9512_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9513_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9514_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(bit_or_not_num_neg(one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9515_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,N)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9516_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,N)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9517_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,N)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9518_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9519_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,N)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9520_cis__mult,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(cis,A2)),aa_real_complex(cis,B2)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ).

% cis_mult
tff(fact_9521_cis__divide,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(cis,A2)),aa_real_complex(cis,B2)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) ).

% cis_divide
tff(fact_9522_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9523_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,N)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9524_or__not__num__neg_Oelims,axiom,
    ! [X2: num,Xa: num,Y3: num] :
      ( ( aa_num_num(bit_or_not_num_neg(X2),Xa) = Y3 )
     => ( ( ( X2 = one )
         => ( ( Xa = one )
           => ( Y3 != one ) ) )
       => ( ( ( X2 = one )
           => ! [M5: num] :
                ( ( Xa = aa_num_num(bit0,M5) )
               => ( Y3 != aa_num_num(bit1,M5) ) ) )
         => ( ( ( X2 = one )
             => ! [M5: num] :
                  ( ( Xa = aa_num_num(bit1,M5) )
                 => ( Y3 != aa_num_num(bit1,M5) ) ) )
           => ( ( ? [N3: num] : X2 = aa_num_num(bit0,N3)
               => ( ( Xa = one )
                 => ( Y3 != aa_num_num(bit0,one) ) ) )
             => ( ! [N3: num] :
                    ( ( X2 = aa_num_num(bit0,N3) )
                   => ! [M5: num] :
                        ( ( Xa = aa_num_num(bit0,M5) )
                       => ( Y3 != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N3),M5)) ) ) )
               => ( ! [N3: num] :
                      ( ( X2 = aa_num_num(bit0,N3) )
                     => ! [M5: num] :
                          ( ( Xa = aa_num_num(bit1,M5) )
                         => ( Y3 != aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N3),M5)) ) ) )
                 => ( ( ? [N3: num] : X2 = aa_num_num(bit1,N3)
                     => ( ( Xa = one )
                       => ( Y3 != one ) ) )
                   => ( ! [N3: num] :
                          ( ( X2 = aa_num_num(bit1,N3) )
                         => ! [M5: num] :
                              ( ( Xa = aa_num_num(bit0,M5) )
                             => ( Y3 != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N3),M5)) ) ) )
                     => ~ ! [N3: num] :
                            ( ( X2 = aa_num_num(bit1,N3) )
                           => ! [M5: num] :
                                ( ( Xa = aa_num_num(bit1,M5) )
                               => ( Y3 != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N3),M5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9525_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_ahl(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_gk,N)))) ) ).

% bij_betw_roots_unity
tff(fact_9526_set__decode__0,axiom,
    ! [X2: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),nat_set_decode(X2)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2)) ) ).

% set_decode_0
tff(fact_9527_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size(3)
tff(fact_9528_set__decode__Suc,axiom,
    ! [N: nat,X2: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),nat_set_decode(X2)))
    <=> pp(aa_set_nat_bool(member_nat(N),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_9529_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_9530_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),nat_set_decode(Z)))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9531_set__decode__def,axiom,
    ! [X2: nat] : nat_set_decode(X2) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ahm,X2)) ).

% set_decode_def
tff(fact_9532_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size_gen(1)
tff(fact_9533_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_9534_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q3),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_9535_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_9536_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_9537_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9538_or__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9539_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_set_complex_bool(bij_be1856998921033663316omplex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C2)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C2)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_gk,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_ahn(C2),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9540_and__int_Oelims,axiom,
    ! [X2: int,Xa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Xa) = Y3 )
     => ( ( ( pp(aa_set_int_bool(member_int(X2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y3 = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_9541_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_9542_real__root__Suc__0,axiom,
    ! [X2: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X2) = X2 ).

% real_root_Suc_0
tff(fact_9543_real__root__eq__iff,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = aa_real_real(aa_nat_fun_real_real(root,N),Y3) )
      <=> ( X2 = Y3 ) ) ) ).

% real_root_eq_iff
tff(fact_9544_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_9545_real__root__eq__0__iff,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = zero_zero_real )
      <=> ( X2 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9546_real__root__less__iff,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3)) ) ) ).

% real_root_less_iff
tff(fact_9547_real__root__le__iff,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) ) ) ).

% real_root_le_iff
tff(fact_9548_real__root__eq__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = one_one_real )
      <=> ( X2 = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9549_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9550_real__root__gt__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) ) ).

% real_root_gt_0_iff
tff(fact_9551_real__root__lt__0__iff,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_9552_real__root__le__0__iff,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_9553_real__root__ge__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3)) ) ) ).

% real_root_ge_0_iff
tff(fact_9554_real__root__lt__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_9555_real__root__gt__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y3)) ) ) ).

% real_root_gt_1_iff
tff(fact_9556_real__root__ge__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y3)) ) ) ).

% real_root_ge_1_iff
tff(fact_9557_real__root__le__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_9558_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9559_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9560_real__root__pow__pos2,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),N) = X2 ) ) ) ).

% real_root_pow_pos2
tff(fact_9561_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9562_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9563_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X2: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),X2) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X2)) ).

% real_root_mult_exp
tff(fact_9564_real__root__pos__pos__le,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ).

% real_root_pos_pos_le
tff(fact_9565_AND__upper2_H,axiom,
    ! [Y3: int,Z: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Y3)),Z)) ) ) ).

% AND_upper2'
tff(fact_9566_AND__upper1_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_9567_AND__upper2,axiom,
    ! [Y3: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Y3)),Y3)) ) ).

% AND_upper2
tff(fact_9568_AND__upper1,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Y3)),X2)) ) ).

% AND_upper1
tff(fact_9569_AND__lower,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Y3))) ) ).

% AND_lower
tff(fact_9570_plus__and__or,axiom,
    ! [X2: int,Y3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Y3)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X2),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Y3) ).

% plus_and_or
tff(fact_9571_real__root__less__mono,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y3))) ) ) ).

% real_root_less_mono
tff(fact_9572_real__root__le__mono,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y3))) ) ) ).

% real_root_le_mono
tff(fact_9573_real__root__power,axiom,
    ! [N: nat,X2: real,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),K) ) ) ).

% real_root_power
tff(fact_9574_real__root__abs,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X2)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)) ) ) ).

% real_root_abs
tff(fact_9575_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K)) ) ).

% and_less_eq
tff(fact_9576_AND__upper1_H_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y3),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_9577_AND__upper2_H_H,axiom,
    ! [Y3: int,Z: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Y3)),Z)) ) ) ).

% AND_upper2''
tff(fact_9578_real__root__gt__zero,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ).

% real_root_gt_zero
tff(fact_9579_real__root__strict__decreasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N6),X2)),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9580_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9581_root__abs__power,axiom,
    ! [N: nat,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N))) = aa_real_real(abs_abs_real,Y3) ) ) ).

% root_abs_power
tff(fact_9582_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)) ) ) ).

% even_and_iff_int
tff(fact_9583_real__root__pos__pos,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ).

% real_root_pos_pos
tff(fact_9584_real__root__strict__increasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N6),X2))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9585_real__root__decreasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N6),X2)),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ) ).

% real_root_decreasing
tff(fact_9586_odd__real__root__pow,axiom,
    ! [N: nat,X2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),N) = X2 ) ) ).

% odd_real_root_pow
tff(fact_9587_odd__real__root__unique,axiom,
    ! [N: nat,Y3: real,X2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N) = X2 )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = Y3 ) ) ) ).

% odd_real_root_unique
tff(fact_9588_odd__real__root__power__cancel,axiom,
    ! [N: nat,X2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) = X2 ) ) ).

% odd_real_root_power_cancel
tff(fact_9589_real__root__pow__pos,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),N) = X2 ) ) ) ).

% real_root_pow_pos
tff(fact_9590_real__root__pos__unique,axiom,
    ! [N: nat,Y3: real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N) = X2 )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = Y3 ) ) ) ) ).

% real_root_pos_unique
tff(fact_9591_real__root__power__cancel,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N)) = X2 ) ) ) ).

% real_root_power_cancel
tff(fact_9592_real__root__increasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N6),X2))) ) ) ) ) ).

% real_root_increasing
tff(fact_9593_log__root,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_nat_fun_real_real(root,N),A2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),A2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_9594_log__base__root,axiom,
    ! [N: nat,B2: real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N),B2)),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),X2)) ) ) ) ).

% log_base_root
tff(fact_9595_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B2))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_9596_and__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_9597_root__powr__inverse,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_9598_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = zero_zero_int ) )
      & ( ~ ( ( K = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = L ) )
          & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_9599_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K),L)))
     => ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_9600_and__int_Opelims,axiom,
    ! [X2: int,Xa: int,Y3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X2),Xa) = Y3 )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Xa)))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y3 = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Xa))) ) ) ) ).

% and_int.pelims
tff(fact_9601_forall__finite_I3_J,axiom,
    ! [X2: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,aa_nat_nat(suc,X2))))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,X2)))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% forall_finite(3)
tff(fact_9602_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9603_and__nat__numerals_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9604_and__nat__numerals_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9605_and__nat__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9606_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9607_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9608_forall__finite_I1_J,axiom,
    ! [P: fun_nat_bool,I5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),zero_zero_nat))
     => pp(aa_nat_bool(P,I5)) ) ).

% forall_finite(1)
tff(fact_9609_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_9610_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9611_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1)))
     => ( ! [K2: int,L2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K2),L2)))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_9612_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X2: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,X2)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),X2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_9613_forall__finite_I2_J,axiom,
    ! [P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,zero_zero_nat)))
         => pp(aa_nat_bool(P,I4)) )
    <=> pp(aa_nat_bool(P,zero_zero_nat)) ) ).

% forall_finite(2)
tff(fact_9614_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1)))
     => ( ! [I2: int,J2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J2)))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_9615_uint32_Osize__eq,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) ).

% uint32.size_eq
tff(fact_9616_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_9617_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_9618_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_9619_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9620_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9621_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q3))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9622_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_9623_num__induct,axiom,
    ! [P: fun_num_bool,X2: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X4: num] :
            ( pp(aa_num_bool(P,X4))
           => pp(aa_num_bool(P,aa_num_num(inc,X4))) )
       => pp(aa_num_bool(P,X2)) ) ) ).

% num_induct
tff(fact_9624_add__inc,axiom,
    ! [X2: num,Y3: num] : aa_num_num(plus_plus_num(X2),aa_num_num(inc,Y3)) = aa_num_num(inc,aa_num_num(plus_plus_num(X2),Y3)) ).

% add_inc
tff(fact_9625_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9626_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ).

% take_bit_nonnegative
tff(fact_9627_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9628_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9629_inc_Osimps_I3_J,axiom,
    ! [X2: num] : aa_num_num(inc,aa_num_num(bit1,X2)) = aa_num_num(bit0,aa_num_num(inc,X2)) ).

% inc.simps(3)
tff(fact_9630_inc_Osimps_I2_J,axiom,
    ! [X2: num] : aa_num_num(inc,aa_num_num(bit0,X2)) = aa_num_num(bit1,X2) ).

% inc.simps(2)
tff(fact_9631_add__One,axiom,
    ! [X2: num] : aa_num_num(plus_plus_num(X2),one) = aa_num_num(inc,X2) ).

% add_One
tff(fact_9632_inc__BitM__eq,axiom,
    ! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_9633_BitM__inc__eq,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_9634_mult__inc,axiom,
    ! [X2: num,Y3: num] : aa_num_num(times_times_num(X2),aa_num_num(inc,Y3)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(X2),Y3)),X2) ).

% mult_inc
tff(fact_9635_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != zero_zero_int )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9636_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9637_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_9638_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9639_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% take_bit_nat_def
tff(fact_9640_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9641_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_9642_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_int_def
tff(fact_9643_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9644_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_9645_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9646_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_9647_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9648_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9649_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9650_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_9651_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9652_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_9653_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_9654_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_aho(L),Qr) ).

% divmod_step_nat_def
tff(fact_9655_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9656_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ahp(L),Qr) ).

% divmod_step_int_def
tff(fact_9657_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9658_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ahq(L),Qr) ).

% divmod_step_integer_def
tff(fact_9659_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = produc2626176000494625587at_nat(aTP_Lamm_ahr,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_9660_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_9661_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_9662_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_nonnegative_int
tff(fact_9663_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_9664_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9665_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9666_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_9667_less__eq__integer__code_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(1)
tff(fact_9668_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_ahs,Qr) ).

% Divides.adjust_div_def
tff(fact_9669_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_9670_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_9671_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% mask_half_int
tff(fact_9672_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% mask_int_def
tff(fact_9673_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_9674_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% divmod_nat_def
tff(fact_9675_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9676_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(K),zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9677_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),L) = L ).

% plus_integer_code(2)
tff(fact_9678_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(K),zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9679_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9680_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9681_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_aht(D2))) ).

% int_ge_less_than_def
tff(fact_9682_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_9683_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X2)) ) ).

% less_eq_integer.abs_eq
tff(fact_9684_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X2)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),X2)) ).

% plus_integer.abs_eq
tff(fact_9685_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X2)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),X2)) ).

% minus_integer.abs_eq
tff(fact_9686_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ahu(D2))) ).

% int_ge_less_than2_def
tff(fact_9687_VEBT__internal_Ospace_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9688_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X2: nat,Y3: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = one_one_int )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = one_one_int )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_9689_vebt__buildup_Opelims,axiom,
    ! [X2: nat,Y3: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_9690_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9691_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_9692_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat)))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
                       => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_9693_VEBT__internal_OTb_Opelims,axiom,
    ! [X2: nat,Y3: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_9694_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_9695_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = one_one_real )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_9696_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X2) = Y3 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = aa_nat_nat(suc,zero_zero_nat) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y3 = aa_nat_nat(suc,zero_zero_nat) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( X2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                       => ( Y3 = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_9697_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = one_one_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_9698_vebt__minti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y3: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(X2) = Y3 )
     => ( pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = vEBT_Leafi(A3,B3) )
             => ( ( ( pp(A3)
                   => ( Y3 = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                  & ( ~ pp(A3)
                   => ( ( pp(B3)
                       => ( Y3 = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                      & ( ~ pp(B3)
                       => ( Y3 = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) )
               => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Leafi(A3,B3))) ) )
         => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y3 = heap_T3487192422709364219on_nat(some_nat(Mi2)) )
                   => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_9699_vebt__maxti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y3: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(X2) = Y3 )
     => ( pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = vEBT_Leafi(A3,B3) )
             => ( ( ( pp(B3)
                   => ( Y3 = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                  & ( ~ pp(B3)
                   => ( ( pp(A3)
                       => ( Y3 = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                      & ( ~ pp(A3)
                       => ( Y3 = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) )
               => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Leafi(A3,B3))) ) )
         => ( ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y3 = heap_T3487192422709364219on_nat(some_nat(Ma2)) )
                   => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_9700_vebt__maxt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_maxt(X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( ( pp(B3)
                   => ( Y3 = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( ( pp(A3)
                       => ( Y3 = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y3 = some_nat(Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9701_vebt__mint_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( vEBT_vebt_mint(X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( ( pp(A3)
                   => ( Y3 = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A3)
                   => ( ( pp(B3)
                       => ( Y3 = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y3 = some_nat(Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9702_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A3),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9703_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B3),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y3 = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y3 = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9704_VEBT__internal_OminNulli_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y3: heap_Time_Heap_bool] :
      ( ( vEBT_VEBT_minNulli(X2) = Y3 )
     => ( pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),X2))
       => ( ( ( X2 = vEBT_Leafi(fFalse,fFalse) )
           => ( ( Y3 = heap_T5885414692442464636n_bool(fTrue) )
             => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi(fFalse,fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X2 = vEBT_Leafi(fTrue,Uv2) )
               => ( ( Y3 = heap_T5885414692442464636n_bool(fFalse) )
                 => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi(fTrue,Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X2 = vEBT_Leafi(Uu2,fTrue) )
                 => ( ( Y3 = heap_T5885414692442464636n_bool(fFalse) )
                   => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi(Uu2,fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( ( X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( ( Y3 = heap_T5885414692442464636n_bool(fTrue) )
                     => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                      ( ( X2 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) )
                     => ( ( Y3 = heap_T5885414692442464636n_bool(fFalse) )
                       => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
tff(fact_9705_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),X2))
       => ( ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
           => ( ( Y3 = one_one_nat )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2) )
               => ( ( Y3 = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue) )
                 => ( ( Y3 = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( ( Y3 = one_one_nat )
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) )
                     => ( ( Y3 = one_one_nat )
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2))) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9706_or__not__num__neg_Opelims,axiom,
    ! [X2: num,Xa: num,Y3: num] :
      ( ( aa_num_num(bit_or_not_num_neg(X2),Xa) = Y3 )
     => ( pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(X2,Xa)))
       => ( ( ( X2 = one )
           => ( ( Xa = one )
             => ( ( Y3 = one )
               => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,one))) ) ) )
         => ( ( ( X2 = one )
             => ! [M5: num] :
                  ( ( Xa = aa_num_num(bit0,M5) )
                 => ( ( Y3 = aa_num_num(bit1,M5) )
                   => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit0,M5)))) ) ) )
           => ( ( ( X2 = one )
               => ! [M5: num] :
                    ( ( Xa = aa_num_num(bit1,M5) )
                   => ( ( Y3 = aa_num_num(bit1,M5) )
                     => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit1,M5)))) ) ) )
             => ( ! [N3: num] :
                    ( ( X2 = aa_num_num(bit0,N3) )
                   => ( ( Xa = one )
                     => ( ( Y3 = aa_num_num(bit0,one) )
                       => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N3),one))) ) ) )
               => ( ! [N3: num] :
                      ( ( X2 = aa_num_num(bit0,N3) )
                     => ! [M5: num] :
                          ( ( Xa = aa_num_num(bit0,M5) )
                         => ( ( Y3 = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N3),M5)) )
                           => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N3),aa_num_num(bit0,M5)))) ) ) )
                 => ( ! [N3: num] :
                        ( ( X2 = aa_num_num(bit0,N3) )
                       => ! [M5: num] :
                            ( ( Xa = aa_num_num(bit1,M5) )
                           => ( ( Y3 = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N3),M5)) )
                             => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N3),aa_num_num(bit1,M5)))) ) ) )
                   => ( ! [N3: num] :
                          ( ( X2 = aa_num_num(bit1,N3) )
                         => ( ( Xa = one )
                           => ( ( Y3 = one )
                             => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N3),one))) ) ) )
                     => ( ! [N3: num] :
                            ( ( X2 = aa_num_num(bit1,N3) )
                           => ! [M5: num] :
                                ( ( Xa = aa_num_num(bit0,M5) )
                               => ( ( Y3 = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N3),M5)) )
                                 => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N3),aa_num_num(bit0,M5)))) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X2 = aa_num_num(bit1,N3) )
                             => ! [M5: num] :
                                  ( ( Xa = aa_num_num(bit1,M5) )
                                 => ( ( Y3 = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N3),M5)) )
                                   => ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N3),aa_num_num(bit1,M5)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9707_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y3: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
      <=> pp(Y3) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X2))
       => ( ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
           => ( pp(Y3)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2) )
               => ( ~ pp(Y3)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue) )
                 => ( ~ pp(Y3)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( pp(Y3)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) )
                     => ( ~ pp(Y3)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9708_arctan__def,axiom,
    ! [Y3: real] : aa_real_real(arctan,Y3) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ahv,Y3)) ).

% arctan_def
tff(fact_9709_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_9710_ln__real__def,axiom,
    ! [X2: real] : aa_real_real(ln_ln_real,X2) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ahw,X2)) ).

% ln_real_def
tff(fact_9711_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9712_ln__neg__is__const,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X2) = the_real(aTP_Lamm_ahx) ) ) ).

% ln_neg_is_const
tff(fact_9713_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_9714_arccos__def,axiom,
    ! [Y3: real] : aa_real_real(arccos,Y3) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ahy,Y3)) ).

% arccos_def
tff(fact_9715_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2))) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9716_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9717_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9718_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4)))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9719_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_ahz) ).

% pi_half
tff(fact_9720_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_ahz)) ).

% pi_def
tff(fact_9721_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X2))
       => ( ! [Uv2: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2))) )
         => ( ! [Uu2: bool] :
                ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue))) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9722_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X2))
       => ( ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
           => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9723_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_9724_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_9725_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_9726_arcsin__def,axiom,
    ! [Y3: real] : aa_real_real(arcsin,Y3) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aia,Y3)) ).

% arcsin_def
tff(fact_9727_zero__le__sgn__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2)) ) ).

% zero_le_sgn_iff
tff(fact_9728_sgn__le__0__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_9729_sgn__root,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)) = aa_real_real(sgn_sgn_real,X2) ) ) ).

% sgn_root
tff(fact_9730_sgn__real__def,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A2) = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
           => ( aa_real_real(sgn_sgn_real,A2) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A2))
           => ( aa_real_real(sgn_sgn_real,A2) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_9731_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_9732_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N)) = X2 )
     => ( ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B2)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_9733_sgn__power__root,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X2))),N)) = X2 ) ) ).

% sgn_power_root
tff(fact_9734_root__sgn__power,axiom,
    ! [N: nat,Y3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y3)),N))) = Y3 ) ) ).

% root_sgn_power
tff(fact_9735_cis__Arg__unique,axiom,
    ! [Z: complex,X2: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
         => ( aa_complex_real(arg,Z) = X2 ) ) ) ) ).

% cis_Arg_unique
tff(fact_9736_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X2: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X2)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N)) = X2 )
             => pp(aa_real_bool(P,Y)) ) ) ) ) ).

% split_root
tff(fact_9737_floor__real__def,axiom,
    ! [X2: real] : aa_real_int(archim6058952711729229775r_real,X2) = the_int(aa_real_fun_int_bool(aTP_Lamm_aib,X2)) ).

% floor_real_def
tff(fact_9738_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ) ).

% Arg_correct
tff(fact_9739_arctan__inverse,axiom,
    ! [X2: real] :
      ( ( X2 != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X2)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X2)) ) ) ).

% arctan_inverse
tff(fact_9740_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9741_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9742_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9743_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9744_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_9745_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_9746_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),N))
      <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9747_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),N))
      <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9748_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% bit_not_int_iff'
tff(fact_9749_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9750_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),M3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M3))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N3)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N3),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N3)) ) ) ) ).

% int_bit_bound
tff(fact_9751_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% bit_int_def
tff(fact_9752_Bit__Operations_Oset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% Bit_Operations.set_bit_eq
tff(fact_9753_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% unset_bit_eq
tff(fact_9754_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_9755_floor__rat__def,axiom,
    ! [X2: rat] : aa_rat_int(archim3151403230148437115or_rat,X2) = the_int(aTP_Lamm_aic(X2)) ).

% floor_rat_def
tff(fact_9756_root__def,axiom,
    ! [N: nat,X2: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_aid,N)),X2) ) ) ) ).

% root_def
tff(fact_9757_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_real(arg,Z) = zero_zero_real ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_real(arg,Z) = fChoice_real(aTP_Lamm_aie(Z)) ) ) ) ).

% Arg_def
tff(fact_9758_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_9759_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_9760_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9761_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9762_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A2) = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
           => ( aa_rat_rat(sgn_sgn_rat,A2) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A2))
           => ( aa_rat_rat(sgn_sgn_rat,A2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9763_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_9764_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9765_less__eq__rat__def,axiom,
    ! [X2: rat,Y3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y3))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y3))
        | ( X2 = Y3 ) ) ) ).

% less_eq_rat_def
tff(fact_9766_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S3))
           => ! [T7: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T7))
               => ( R2 != aa_rat_rat(plus_plus_rat(S3),T7) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9767_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_9768_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9769_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9770_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9771_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9772_even__not__iff__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_not_iff_int
tff(fact_9773_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_9774_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9775_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(2)
tff(fact_9776_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9777_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_9778_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),N))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N)) ) ).

% bit_minus_int_iff
tff(fact_9779_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(5)
tff(fact_9780_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_9781_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(3)
tff(fact_9782_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9783_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9784_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ) ).

% bit_nat_def
tff(fact_9785_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(9)
tff(fact_9786_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(6)
tff(fact_9787_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% or_not_numerals(6)
tff(fact_9788_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_9789_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% and_not_numerals(8)
tff(fact_9790_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_9791_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_9792_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9793_int__not__code_I1_J,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_9794_bitNOT__integer__code,axiom,
    ! [I: code_integer] : aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,I) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,I)),one_one_Code_integer) ).

% bitNOT_integer_code
tff(fact_9795_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_9796_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9797_XOR__lower,axiom,
    ! [X2: int,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X2),Y3))) ) ) ).

% XOR_lower
tff(fact_9798_diff__rat__def,axiom,
    ! [Q3: rat,R2: rat] : aa_rat_rat(minus_minus_rat(Q3),R2) = aa_rat_rat(plus_plus_rat(Q3),aa_rat_rat(uminus_uminus_rat,R2)) ).

% diff_rat_def
tff(fact_9799_XOR__upper,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X2),Y3)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_9800_xor__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9801_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: bool] : bits_Bit_integer(aa_int_Code_integer(code_integer_of_int,Xa),X2) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,X2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))) ).

% Bit_integer.abs_eq
tff(fact_9802_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3)) ).

% xor_nat_numerals(1)
tff(fact_9803_xor__nat__numerals_I2_J,axiom,
    ! [Y3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y3))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y3)) ).

% xor_nat_numerals(2)
tff(fact_9804_xor__nat__numerals_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2)) ).

% xor_nat_numerals(3)
tff(fact_9805_xor__nat__numerals_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2)) ).

% xor_nat_numerals(4)
tff(fact_9806_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_9807_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9808_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_9809_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_9810_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9811_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9812_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_9813_dup__1,axiom,
    aa_Cod1783974652533396728nteger(code_dup,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% dup_1
tff(fact_9814_pow_Osimps_I1_J,axiom,
    ! [X2: num] : aa_num_num(pow(X2),one) = X2 ).

% pow.simps(1)
tff(fact_9815_dup_Oabs__eq,axiom,
    ! [X2: int] : aa_Cod1783974652533396728nteger(code_dup,aa_int_Code_integer(code_integer_of_int,X2)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),X2)) ).

% dup.abs_eq
tff(fact_9816_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9817_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9818_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9819_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = produc4245557441103728435nt_int(aTP_Lamm_aif,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9820_setceilmax,axiom,
    ! [S2: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,S2),M))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Listy)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
       => ( ( M = aa_nat_nat(suc,N) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Listy)))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X4)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S2,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9821_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T),set_VEBT_VEBT2(TreeList)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList)))))) ) ).

% height_compose_list
tff(fact_9822_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),lattic8265883725875713057ax_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ).

% max_ins_scaled
tff(fact_9823_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(X13),I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))) ) ).

% height_i_max
tff(fact_9824_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(X13),I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))))) ) ).

% max_idx_list
tff(fact_9825_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_du,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9826_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9827_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9828_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9829_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9830_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9831_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9832_VEBT__internal_Oheight_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X2) = Y3 )
     => ( ( ? [A3: bool,B3: bool] : X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3)
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) )
             => ( Y3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9833_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aig(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9834_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = produc4245557441103728435nt_int(aTP_Lamm_aih,quotient_of(P2)) ).

% rat_abs_code
tff(fact_9835_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = produc4245557441103728435nt_int(aTP_Lamm_aii,quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9836_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),Q3))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_aik(Q3)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_9837_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P2)) ).

% rat_floor_code
tff(fact_9838_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P2),Q3))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_aim(Q3)),quotient_of(P2))) ) ).

% rat_less_eq_code
tff(fact_9839_VEBT__internal_Oheight_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X2) = Y3 )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),X2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3) )
             => ( ( Y3 = zero_zero_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B3))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) )
               => ( ( Y3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9840_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(aa_int_rat(of_int,A2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(A2),one_one_int) ).

% quotient_of_int
tff(fact_9841_bij__betw__Suc,axiom,
    ! [M8: set_nat,N6: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M8),N6))
    <=> ( image_nat_nat(suc,M8) = N6 ) ) ).

% bij_betw_Suc
tff(fact_9842_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9843_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9844_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_dv,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9845_zero__notin__Suc__image,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A))) ).

% zero_notin_Suc_image
tff(fact_9846_finite__int__iff__bounded,axiom,
    ! [S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S)),set_ord_lessThan_int(K3))) ) ).

% finite_int_iff_bounded
tff(fact_9847_finite__int__iff__bounded__le,axiom,
    ! [S: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S)),set_ord_atMost_int(K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_9848_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = set_or1269000886237332187st_nat(one_one_nat,N) ).

% image_Suc_lessThan
tff(fact_9849_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9850_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9851_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9852_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9853_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_9854_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ain,N),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N) ) ) ).

% range_mod
tff(fact_9855_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_aio,L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9856_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_aip(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9857_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9858_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y3: nat,X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),Y3))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq,C2),set_or4665077453230672383an_nat(X2,Y3)) = set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y3),C2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),Y3))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq,C2),set_or4665077453230672383an_nat(X2,Y3)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y3))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq,C2),set_or4665077453230672383an_nat(X2,Y3)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9859_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_ais(Q3),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9860_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_aiu(Q3),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9861_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P2),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9862_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_aiw(Q3),quotient_of(P2)) ).

% rat_times_code
tff(fact_9863_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_aiy(Q3),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9864_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_9865_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9866_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9867_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9868_bin__rest__integer_Oabs__eq,axiom,
    ! [X2: int] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,aa_int_Code_integer(code_integer_of_int,X2)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9869_bin__rest__integer__code,axiom,
    ! [I: code_integer] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,I) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% bin_rest_integer_code
tff(fact_9870_bitXOR__integer__unfold,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( ( X2 = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(X2),Y3) = Y3 ) )
      & ( ( X2 != zero_z3403309356797280102nteger )
       => ( ( ( X2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(X2),Y3) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,Y3) ) )
          & ( ( X2 != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(X2),Y3) = bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,X2)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y3)),aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_Code_integer_bool(bits_b8758750999018896077nteger,X2))),aa_Code_integer_bool(bits_b8758750999018896077nteger,Y3))) ) ) ) ) ) ).

% bitXOR_integer_unfold
tff(fact_9871_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9872_drop__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9873_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9874_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9875_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9876_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9877_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9878_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,N)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_9879_bitval__bin__last__integer,axiom,
    ! [I: code_integer] : zero_n356916108424825756nteger(aa_Code_integer_bool(bits_b8758750999018896077nteger,I)) = aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(I),one_one_Code_integer) ).

% bitval_bin_last_integer
tff(fact_9880_shiftr__integer__conv__div__pow2,axiom,
    ! [N: nat,X2: code_integer] : aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),X2) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9881_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( pp(aa_Code_integer_bool(bits_b8758750999018896077nteger,I))
    <=> ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(I),one_one_Code_integer) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_code
tff(fact_9882_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,one_one_nat),I) ).

% bin_rest_code
tff(fact_9883_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% drop_bit_int_def
tff(fact_9884_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% drop_bit_nat_def
tff(fact_9885_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( pp(aa_Code_integer_bool(bits_b8758750999018896077nteger,I))
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9886_bin__last__integer_Oabs__eq,axiom,
    ! [X2: int] :
      ( pp(aa_Code_integer_bool(bits_b8758750999018896077nteger,aa_int_Code_integer(code_integer_of_int,X2)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X2)) ) ).

% bin_last_integer.abs_eq
tff(fact_9887_bitOR__integer__unfold,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( ( X2 = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(X2),Y3) = Y3 ) )
      & ( ( X2 != zero_z3403309356797280102nteger )
       => ( ( ( X2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(X2),Y3) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ( X2 != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(X2),Y3) = bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,X2)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y3)),aa_bool_bool(fdisj(aa_Code_integer_bool(bits_b8758750999018896077nteger,X2)),aa_Code_integer_bool(bits_b8758750999018896077nteger,Y3))) ) ) ) ) ) ).

% bitOR_integer_unfold
tff(fact_9888_bitAND__integer__unfold,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( ( X2 = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(X2),Y3) = zero_z3403309356797280102nteger ) )
      & ( ( X2 != zero_z3403309356797280102nteger )
       => ( ( ( X2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(X2),Y3) = Y3 ) )
          & ( ( X2 != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(X2),Y3) = bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,X2)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y3)),aa_bool_bool(fconj(aa_Code_integer_bool(bits_b8758750999018896077nteger,X2)),aa_Code_integer_bool(bits_b8758750999018896077nteger,Y3))) ) ) ) ) ) ).

% bitAND_integer_unfold
tff(fact_9889_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9890_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_9891_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9892_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9893_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9894_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,fFalse) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(one_one_nat),I) ).

% Bit_integer_code(1)
tff(fact_9895_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9896_Bit__Operations_Oset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9897_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q3)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9898_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% flip_bit_int_def
tff(fact_9899_shiftl__integer__conv__mult__pow2,axiom,
    ! [N: nat,X2: code_integer] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),X2) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9900_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ).

% unset_bit_int_def
tff(fact_9901_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_int_def
tff(fact_9902_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,fTrue) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(one_one_nat),I)),one_one_Code_integer) ).

% Bit_integer_code(2)
tff(fact_9903_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_9904_push__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_minus_one
tff(fact_9905_set__bit__integer__conv__masks,axiom,
    ! [B2: bool,X2: code_integer,I: nat] :
      ( ( pp(B2)
       => ( generi2397576812484419408nteger(X2,I,B2) = aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(X2),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(I),one_one_Code_integer)) ) )
      & ( ~ pp(B2)
       => ( generi2397576812484419408nteger(X2,I,B2) = aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(X2),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(I),one_one_Code_integer))) ) ) ) ).

% set_bit_integer_conv_masks
tff(fact_9906_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N: nat] : generi8991105624351003935it_int(I,N,fTrue) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).

% int_set_bit_True_conv_OR
tff(fact_9907_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N: nat] : generi8991105624351003935it_int(I,N,fFalse) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ).

% int_set_bit_False_conv_NAND
tff(fact_9908_int__set__bit__conv__ops,axiom,
    ! [B2: bool,I: int,N: nat] :
      ( ( pp(B2)
       => ( generi8991105624351003935it_int(I,N,B2) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ) )
      & ( ~ pp(B2)
       => ( generi8991105624351003935it_int(I,N,B2) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ) ) ) ).

% int_set_bit_conv_ops
tff(fact_9909_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_9910_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,N)),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_9911_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_9912_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),R2) ).

% concat_bit_assoc
tff(fact_9913_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),L)) ).

% concat_bit_eq
tff(fact_9914_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9915_int__lsb__numeral_I2_J,axiom,
    pp(aa_int_bool(least_4859182151741483524sb_int,one_one_int)) ).

% int_lsb_numeral(2)
tff(fact_9916_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ pp(aa_int_bool(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))) ).

% int_lsb_numeral(6)
tff(fact_9917_int__lsb__numeral_I3_J,axiom,
    pp(aa_int_bool(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(3)
tff(fact_9918_int__lsb__numeral_I4_J,axiom,
    pp(aa_int_bool(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,one_one_int))) ).

% int_lsb_numeral(4)
tff(fact_9919_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ pp(aa_int_bool(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))))) ).

% int_lsb_numeral(8)
tff(fact_9920_int__lsb__numeral_I5_J,axiom,
    pp(aa_int_bool(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)))) ).

% int_lsb_numeral(5)
tff(fact_9921_bin__last__conv__lsb,axiom,
    aTP_Lamm_aiz = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9922_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_9923_Cauchy__iff2,axiom,
    ! [X: fun_nat_real] :
      ( topolo4055970368930404560y_real(X)
    <=> ! [J3: nat] :
        ? [M10: nat] :
        ! [M7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),M7))
         => ! [N7: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),N7))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X,M7)),aa_nat_real(X,N7)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9924_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),B2) = aa_int_int(uminus_uminus_int,A2) )
      <=> ( B2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_9925_int__sdiv__simps_I3_J,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A2) ).

% int_sdiv_simps(3)
tff(fact_9926_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),one_one_int) = A2 ).

% int_sdiv_simps(1)
tff(fact_9927_int__sdiv__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),B2) = A2 )
      <=> ( B2 = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_9928_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(fI_QUERY(P),Q),top_top_assn))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(Q),top_top_assn))) ) ).

% entails_solve_init(1)
tff(fact_9929_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F3: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(fI_QUERY(P),Q),F3))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(Q),F3))) ) ).

% frame_inference_init
tff(fact_9930_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F3: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(fI_QUERY(P),Q),F3))
    <=> pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(times_times_assn(Q),F3))) ) ).

% FI_QUERY_def
tff(fact_9931_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(fI_QUERY(P),Q),one_one_assn))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ).

% entails_solve_init(2)
tff(fact_9932_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B2))),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,B2)),one_one_int)))) ) ).

% smod_int_range
tff(fact_9933_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),B2)) ) ) ).

% smod_int_compares(1)
tff(fact_9934_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2))) ) ) ).

% smod_int_compares(2)
tff(fact_9935_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),zero_zero_int)) ) ) ).

% smod_int_compares(4)
tff(fact_9936_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2))) ) ) ).

% smod_int_compares(6)
tff(fact_9937_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),zero_zero_int)) ) ) ).

% smod_int_compares(7)
tff(fact_9938_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2))) ) ) ).

% smod_int_compares(8)
tff(fact_9939_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ) ) ) ).

% smod_mod_positive
tff(fact_9940_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,K),L)),L)) ).

% signed_modulo_int_def
tff(fact_9941_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),aa_int_int(uminus_uminus_int,B2))) ) ) ).

% smod_int_compares(5)
tff(fact_9942_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2))) ) ) ).

% smod_int_compares(3)
tff(fact_9943_uint32_Osize__eq__length,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9944_len__num1,axiom,
    ! [Uu: itself_Numeral_num1] : type_l4264026598287037465l_num1(Uu) = one_one_nat ).

% len_num1
tff(fact_9945_len__of__finite__1__def,axiom,
    ! [X2: itself_finite_1] : type_l31302759751748491nite_1(X2) = one_one_nat ).

% len_of_finite_1_def
tff(fact_9946_len__of__finite__2__def,axiom,
    ! [X2: itself_finite_2] : type_l31302759751748492nite_2(X2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% len_of_finite_2_def
tff(fact_9947_len__of__finite__3__def,axiom,
    ! [X2: itself_finite_3] : type_l31302759751748493nite_3(X2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% len_of_finite_3_def
tff(fact_9948_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_9949_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_9950_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9951_min__minus,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus
tff(fact_9952_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus'
tff(fact_9953_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A2)),B2) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_9954_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),aa_nat_nat(suc,A2)) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_9955_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),A2)) = A2 ).

% rev_min_pm1
tff(fact_9956_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ).

% rev_min_pm
tff(fact_9957_min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)) = A2 ).

% min_pm1
tff(fact_9958_min__pm,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ).

% min_pm
tff(fact_9959_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_9960_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9961_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_9962_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_min_right
tff(fact_9963_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_min_left
tff(fact_9964_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),R2) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9965_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9966_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N))) ).

% mod_mod_power
tff(fact_9967_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat,Q3) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9968_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : ord_mi8085742599997312461d_enat(Q3,zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9969_shiftl__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% shiftl_Suc_0
tff(fact_9970_shiftr__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% shiftr_Suc_0
tff(fact_9971_inj__on__diff__nat,axiom,
    ! [N6: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( pp(aa_set_nat_bool(member_nat(N3),N6))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N3)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq,K)),N6)) ) ).

% inj_on_diff_nat
tff(fact_9972_inj__Suc,axiom,
    ! [N6: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N6)) ).

% inj_Suc
tff(fact_9973_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_aid,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_9974_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D2))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D2)) ) ).

% valid_eq2
tff(fact_9975_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D2))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D2)) ) ).

% valid_eq1
tff(fact_9976_valid__eq,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D2))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D2)) ) ).

% valid_eq
tff(fact_9977_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),D2))
    <=> ( D2 = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9978_DERIV__real__root__generic,axiom,
    ! [N: nat,X2: real,D: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X2 != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
             => ( D = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
               => ( D = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
               => ( D = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D,topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9979_DERIV__even__real__root,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9980_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X2: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H6)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H6)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9981_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X2: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H6)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H6))),aa_real_real(F,X2))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9982_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X2: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H6)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H6)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9983_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X2: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,S))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H6)),S))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H6))),aa_real_real(F,X2))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9984_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X2: real,D2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ( ! [Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y4))),D2))
             => ( aa_real_real(F,X2) = aa_real_real(F,Y4) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9985_MVT2,axiom,
    ! [A2: real,B2: real,F: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B2))
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),aa_real_real(F5,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9986_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y3: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,X2),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_aja(F),aa_real_real(uminus_uminus_real,Y3),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9987_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_or1222579329274155063t_real(A2,B2)))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1222579329274155063t_real(A2,B2)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X4))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(G,A2)),aa_real_real(G,B2))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9988_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y5),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B2)),aa_real_real(F,A2))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9989_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y5)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A2)),aa_real_real(F,B2))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9990_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y5),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B2)),aa_real_real(F,A2))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9991_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y5)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A2)),aa_real_real(F,B2))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9992_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H6)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9993_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H6))),aa_real_real(F,X2))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9994_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H6))),aa_real_real(F,X2))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9995_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H6))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H6),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H6)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9996_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9997_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9998_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A2)),aa_real_real(V,B2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9999_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X2: real,D2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ( ! [Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y4))),D2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),aa_real_real(F,Y4))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_10000_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X2: real,D2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ( ! [Y4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y4))),D2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y4)),aa_real_real(F,X2))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_10001_DERIV__ln__divide,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_10002_DERIV__pow,axiom,
    ! [N: nat,X2: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ajb,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X2,S2)) ).

% DERIV_pow
tff(fact_10003_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X2: real,N: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X2,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ajc(G),N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_10004_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ajd,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_10005_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X2: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X2)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_aje(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_10006_DERIV__log,axiom,
    ! [X2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B2)),X2)),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_log
tff(fact_10007_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X2: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X2)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X2,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ajf(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X2)),aa_real_real(F,X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,X2)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,X2))),aa_real_real(G,X2)))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_10008_DERIV__real__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_10009_DERIV__arctan,axiom,
    ! [X2: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ).

% DERIV_arctan
tff(fact_10010_arsinh__real__has__field__derivative,axiom,
    ! [X2: real,A: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X2,A)) ).

% arsinh_real_has_field_derivative
tff(fact_10011_DERIV__real__sqrt__generic,axiom,
    ! [X2: real,D: real] :
      ( ( X2 != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( D = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
           => ( D = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D,topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_10012_arcosh__real__has__field__derivative,axiom,
    ! [X2: real,A: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X2,A)) ) ).

% arcosh_real_has_field_derivative
tff(fact_10013_artanh__real__has__field__derivative,axiom,
    ! [X2: real,A: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X2,A)) ) ).

% artanh_real_has_field_derivative
tff(fact_10014_DERIV__real__root,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_10015_DERIV__arccos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_10016_DERIV__arcsin,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_10017_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X2: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M5: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
            & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ajg(Diff),X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_10018_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X2: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M5: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T7: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
          & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ajg(Diff),X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_10019_DERIV__odd__real__root,axiom,
    ! [N: nat,X2: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( ( X2 != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_10020_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M5: nat,T7: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
           => ? [T7: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ajh(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_10021_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
         => ? [T7: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T7))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ajh(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10022_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M5: nat,T7: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T7))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
           => ? [T7: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ajh(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10023_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X2: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X2 != zero_zero_real )
         => ( ! [M5: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T7: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T7)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
                & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ajg(Diff),X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10024_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X2: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M5: nat,T7: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
       => ? [T7: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T7)),aa_real_real(abs_abs_real,X2)))
            & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ajg(Diff),X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10025_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A2: real,B2: real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),T7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),B2)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B2))
             => ? [T7: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),T7))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),C2))
                  & ( aa_real_real(F,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aji(Diff),A2),C2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10026_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A2: real,B2: real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),T7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),B2)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B2))
             => ? [T7: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),T7))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),B2))
                  & ( aa_real_real(F,B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aji(Diff),B2),C2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10027_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A2: real,B2: real,C2: real,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),T7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),B2)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B2))
                 => ( ( X2 != C2 )
                   => ? [T7: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C2))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T7))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),C2)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C2))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),T7))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T7),X2)) ) )
                        & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ajj(Diff),C2),X2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T7)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10028_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B: real] :
      ( ! [M5: nat,T7: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T7),topolo2177554685111907308n_real(T7,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M3: nat,T8: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T8))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T8),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ajl(N),Diff),B),M3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T8)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ajm(Diff),M3),T8),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M3))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T8),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M3)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M3))))))),topolo2177554685111907308n_real(T8,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10029_DERIV__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_ajn,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ajo,X2)),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10030_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_ajp(F),X4)) )
     => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
         => has_fi5821293074295781190e_real(aTP_Lamm_ajr(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ajp(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_10031_tanh__real__bounds,axiom,
    ! [X2: real] : pp(aa_set_real_bool(member_real(aa_real_real(tanh_real,X2)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))) ).

% tanh_real_bounds
tff(fact_10032_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F5: fun_re4238438383274954274t_real,X0: real,A2: real,B2: real,L5: fun_nat_real] :
      ( ! [N3: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_vl(F),N3),aa_nat_real(aa_real_fun_nat_real(F5,X0),N3),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(A2,B2)))
           => summable_real(aa_real_fun_nat_real(F,X4)) )
       => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(A2,B2)))
         => ( summable_real(aa_real_fun_nat_real(F5,X0))
           => ( summable_real(L5)
             => ( ! [N3: nat,X4: real,Y4: real] :
                    ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(A2,B2)))
                   => ( pp(aa_set_real_bool(member_real(Y4),set_or1633881224788618240n_real(A2,B2)))
                     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X4),N3)),aa_nat_real(aa_real_fun_nat_real(F,Y4),N3)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N3)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),Y4))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_ajs(F),suminf_real(aa_real_fun_nat_real(F5,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_10033_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_or4266950643985792945nteger(L,U))) ).

% finite_greaterThanLessThan_integer
tff(fact_10034_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_10035_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ? [L6: real,M9: real] :
            ( ! [X6: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X6),B2)) )
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_real_real(F,X6)))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X6)),M9)) ) )
            & ! [Y5: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),Y5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y5),M9)) )
               => ? [X4: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
                    & ( aa_real_real(F,X4) = Y5 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_10036_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X6: real] :
                ( ( ( X6 != C2 )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X6))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X6))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_10037_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X6: real] :
                ( ( ( X6 != C2 )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X6))),R3)) )
               => ( aa_real_real(F,X6) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_10038_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X6: real] :
                ( ( ( X6 != C2 )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X6))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X6)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_10039_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_10040_isCont__inverse__function2,axiom,
    ! [A2: real,X2: real,B2: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B2))
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B2))
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B2))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X2),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_10041_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_10042_isCont__arcosh,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_10043_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_ajt,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10044_isCont__arccos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_10045_isCont__arcsin,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_10046_LIM__less__bound,axiom,
    ! [B2: real,X2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),X2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(B2,X2)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) ) ) ) ).

% LIM_less_bound
tff(fact_10047_isCont__artanh,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_10048_isCont__inverse__function,axiom,
    ! [D2: real,X2: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
     => ( ! [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X2))),D2))
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X2))),D2))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X2),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_10049_GMVT_H,axiom,
    ! [A2: real,B2: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ! [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B2))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),Z2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B2))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B2))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B2))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B2))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2))),aa_real_real(G2,C3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B2)),aa_real_real(G,A2))),aa_real_real(F5,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10050_summable__Leibniz_I2_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A2,zero_zero_nat)))
         => ! [N8: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_aju(A2))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8))),groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10051_summable__Leibniz_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A2,zero_zero_nat)),zero_zero_real))
         => ! [N8: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_aju(A2))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10052_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10053_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_10054_monoseq__convergent,axiom,
    ! [X: fun_nat_real,B: real] :
      ( topolo6980174941875973593q_real(X)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X,I2))),B))
       => ~ ! [L6: real] : ~ filterlim_nat_real(X,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_10055_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_ajv,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10056_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N3))),aa_nat_real(G,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(G,N3)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nf(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N8)),L2))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),aa_nat_real(G,N8)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10057_LIMSEQ__inverse__zero,axiom,
    ! [X: fun_nat_real] :
      ( ! [R3: real] :
        ? [N9: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X,N3))) )
     => filterlim_nat_real(aTP_Lamm_ajw(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10058_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_ajx,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10059_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_ajy,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10060_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajz,C2),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10061_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aka,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10062_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),L))
       => ( ! [E2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
             => ? [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N8)),E2))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10063_LIMSEQ__realpow__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10064_LIMSEQ__divide__realpow__zero,axiom,
    ! [X2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_akb(X2),A2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10065_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10066_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C2)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10067_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_akc,X2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10068_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_akd,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10069_tendsto__exp__limit__sequentially,axiom,
    ! [X2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ake,X2),topolo2815343760600316023s_real(aa_real_real(exp_real,X2)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10070_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_akf,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10071_summable__Leibniz_I1_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => summable_real(aTP_Lamm_aju(A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_10072_summable,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3)))
         => summable_real(aTP_Lamm_aju(A2)) ) ) ) ).

% summable
tff(fact_10073_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_akg(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_akh(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10074_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_aki(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_akh(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10075_summable__Leibniz_I4_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => filterlim_nat_real(aTP_Lamm_akj(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aju(A2))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10076_zeroseq__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahf,X2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10077_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3)))
         => filterlim_nat_real(aTP_Lamm_akj(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aju(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10078_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_aju(A2)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10079_sums__alternating__upper__lower,axiom,
    ! [A2: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3)))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3)))
       => ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)))),L2))
              & filterlim_nat_real(aTP_Lamm_akj(A2),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_akk(A2),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10080_summable__Leibniz_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => filterlim_nat_real(aTP_Lamm_akk(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aju(A2))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10081_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3)))
         => filterlim_nat_real(aTP_Lamm_akk(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_aju(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10082_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_aju(A2))),groups6591440286371151544t_real(aTP_Lamm_aju(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10083_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_10084_dist__complex__def,axiom,
    ! [X2: complex,Y3: complex] : aa_complex_real(real_V3694042436643373181omplex(X2),Y3) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X2),Y3)) ).

% dist_complex_def
tff(fact_10085_dist__real__def,axiom,
    ! [X2: real,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,X2),Y3) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3)) ).

% dist_real_def
tff(fact_10086_tendsto__exp__limit__at__right,axiom,
    ! [X2: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_akl,X2),topolo2815343760600316023s_real(aa_real_real(exp_real,X2)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10087_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10088_artanh__real__at__right__1,axiom,
    filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).

% artanh_real_at_right_1
tff(fact_10089_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_10090_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10091_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_10092_log__inj,axiom,
    ! [B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B2))
     => pp(aa_set_real_bool(inj_on_real_real(aa_rea7340605660300375871l_real(log,B2)),set_or5849166863359141190n_real(zero_zero_real))) ) ).

% log_inj
tff(fact_10093_tanh__real__at__bot,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).

% tanh_real_at_bot
tff(fact_10094_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).

% tendsto_arcosh_at_left_1
tff(fact_10095_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2))
         => ? [Y5: real] :
              ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y5)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B2))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_10096_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akm(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10097_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akm(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10098_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_10099_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_10100_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_10101_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_10102_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_10103_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_10104_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real(uminus_uminus_real,at_bot_real,at_top_real) ).

% filterlim_uminus_at_bot_at_top
tff(fact_10105_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real(uminus_uminus_real,at_top_real,at_bot_real) ).

% filterlim_uminus_at_top_at_bot
tff(fact_10106_greaterThan__0,axiom,
    set_or1210151606488870762an_nat(zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_10107_greaterThan__Suc,axiom,
    ! [K: nat] : set_or1210151606488870762an_nat(aa_nat_nat(suc,K)) = minus_minus_set_nat(set_or1210151606488870762an_nat(K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_10108_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_10109_artanh__real__at__left__1,axiom,
    filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).

% artanh_real_at_left_1
tff(fact_10110_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_akn,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10111_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_10112_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_10113_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_ako,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10114_tendsto__exp__limit__at__top,axiom,
    ! [X2: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_akp,X2),topolo2815343760600316023s_real(aa_real_real(exp_real,X2)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10115_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10116_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10117_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),X4))
         => ? [Y5: real] :
              ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y5),zero_zero_real)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B2))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10118_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X2: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y3: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
     => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G),F),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10119_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_akt(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10120_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_aku(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10121_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_aku(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10122_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N10: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N10),F3) ) ).

% le_sequentially
tff(fact_10123_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N10: nat] :
        ! [N7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N10),N7))
         => pp(aa_nat_bool(P,N7)) ) ) ).

% eventually_sequentially
tff(fact_10124_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X4))
         => pp(aa_nat_bool(P,X4)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10125_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_as,at_top_nat) ).

% eventually_False_sequentially
tff(fact_10126_eventually__at__right__to__0,axiom,
    ! [P: fun_real_bool,A2: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
    <=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_akv(P),A2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10127_eventually__at__left__to__right,axiom,
    ! [P: fun_real_bool,A2: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
    <=> eventually_real(aTP_Lamm_akw(P),topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,A2),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,A2)))) ) ).

% eventually_at_left_to_right
tff(fact_10128_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_akx(A2),B2),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ).

% eventually_at_right_real
tff(fact_10129_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A2))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_akx(B2),A2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ).

% eventually_at_left_real
tff(fact_10130_eventually__at__right__to__top,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_aky(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_10131_eventually__at__top__to__right,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_aky(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_10132_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(A2,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(A2,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F5),G2),at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_10133_lhopital,axiom,
    ! [F: fun_real_real,X2: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( eventually_real(aTP_Lamm_akq(G),topolo2177554685111907308n_real(X2,top_top_set_real))
         => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(X2,top_top_set_real))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(X2,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),F3,topolo2177554685111907308n_real(X2,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),F3,topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10134_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F5),G2),at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_10135_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A2,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(A2,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(A2,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F5),G2),at_bot_real,topolo2177554685111907308n_real(A2,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),at_bot_real,topolo2177554685111907308n_real(A2,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_10136_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F5),G2),at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_10137_lhopital__at__top,axiom,
    ! [G: fun_real_real,X2: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y3: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(X2,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(X2,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G),F),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10138_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,X2: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_akq(G2),at_top_real)
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),at_top_real)
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),topolo2815343760600316023s_real(X2),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G),F),topolo2815343760600316023s_real(X2),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10139_lhopital__right,axiom,
    ! [F: fun_real_real,X2: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
       => ( eventually_real(aTP_Lamm_akq(G),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
         => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),F3,topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),F3,topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10140_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_akq(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F0),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10141_lhopital__left,axiom,
    ! [F: fun_real_real,X2: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
       => ( eventually_real(aTP_Lamm_akq(G),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
         => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),F3,topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),F3,topolo2177554685111907308n_real(X2,set_or5984915006950818249n_real(X2))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10142_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F5),G2),at_bot_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),at_bot_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_10143_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F5),G2),at_bot_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(F),G),at_bot_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_10144_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X2: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y3: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
     => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G),F),topolo2815343760600316023s_real(Y3),topolo2177554685111907308n_real(X2,set_or5849166863359141190n_real(X2))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10145_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,X2: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_akq(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(F),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G2),F5),topolo2815343760600316023s_real(X2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(G),F),topolo2815343760600316023s_real(X2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10146_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A2: real,B2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(image_nat_real(F,top_top_set_nat)),set_or1222579329274155063t_real(A2,B2)))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_10147_Bseq__realpow,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,X2),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10148_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B2: nat] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
     => ( ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),B2)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_10149_Greatest__le__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),B2)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_10150_GreatestI__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y4: nat] :
            ( pp(aa_nat_bool(P,Y4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),B2)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_10151_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_10152_suminf__eq__SUP__real,axiom,
    ! [X: fun_nat_real] :
      ( summable_real(X)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X,I2)))
       => ( suminf_real(X) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_ala(X),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_10153_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_or2715278749043346189nteger(L,U))) ).

% finite_greaterThanAtMost_integer
tff(fact_10154_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_10155_decseq__bounded,axiom,
    ! [X: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_nat_real(X,I2)))
       => bfun_nat_real(X,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_10156_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_10157_decseq__convergent,axiom,
    ! [X: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_nat_real(X,I2)))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_nat_real(X,I5))) ) ) ) ).

% decseq_convergent
tff(fact_10158_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_10159_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_10160_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = set_or1210151606488870762an_nat(K) ).

% atLeast_Suc_greaterThan
tff(fact_10161_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_10162_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_10163_GMVT,axiom,
    ! [A2: real,B2: real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B2)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B2)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B2)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B2))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B2)),aa_real_real(G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10164_MVT,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B2))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L2: real,Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Z2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B2))
              & has_fi5821293074295781190e_real(F,L2,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),L2) ) ) ) ) ) ).

% MVT
tff(fact_10165_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_10166_card__Collect__less__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gm,N))) = N ).

% card_Collect_less_nat
tff(fact_10167_card__atMost,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_10168_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(set_or4665077453230672383an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_10169_card__Collect__le__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gl,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_10170_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(set_or6659071591806873216st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_10171_card__UNIV__bool,axiom,
    finite_card_bool(top_top_set_bool) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_10172_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(set_or1269000886237332187st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_10173_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_10174_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(set_or5834768355832116004an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_10175_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_10176_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_10177_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_10178_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),B2))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F)
       => ? [C3: real,D4: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A2,B2)) = set_or1222579329274155063t_real(C3,D4) )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C3),D4)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_10179_continuous__on__arsinh_H,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F)
     => topolo5044208981011980120l_real(A,aTP_Lamm_alb(F)) ) ).

% continuous_on_arsinh'
tff(fact_10180_card__less__Suc2,axiom,
    ! [M8: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_alc(M8),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ald(M8),I))) ) ) ).

% card_less_Suc2
tff(fact_10181_card__less__Suc,axiom,
    ! [M8: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_alc(M8),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ald(M8),I))) ) ) ).

% card_less_Suc
tff(fact_10182_card__less,axiom,
    ! [M8: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ald(M8),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_10183_subset__card__intvl__is__intvl,axiom,
    ! [A: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A)))))
     => ( A = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_10184_continuous__on__arcosh_H,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F)
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
       => topolo5044208981011980120l_real(A,aTP_Lamm_ale(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10185_continuous__on__arcosh,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),set_ord_atLeast_real(one_one_real)))
     => topolo5044208981011980120l_real(A,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10186_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_10187_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_10188_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N6),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(N6)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_10189_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(S)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S)))) ) ).

% card_le_Suc_Max
tff(fact_10190_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(zero_zero_nat,finite_card_nat(S)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),S))) ).

% card_sum_le_nat_sum
tff(fact_10191_continuous__on__artanh_H,axiom,
    ! [A: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F)
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A))
           => pp(aa_set_real_bool(member_real(aa_real_real(F,X4)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))) )
       => topolo5044208981011980120l_real(A,aTP_Lamm_alf(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10192_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_ahn(C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_10193_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_gk,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_10194_mvt,axiom,
    ! [A2: real,B2: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B2))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ~ ! [Xi3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),Xi3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi3),B2))
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A2)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_10195_continuous__on__artanh,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)))
     => topolo5044208981011980120l_real(A,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10196_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F: fun_real_real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B2))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A2),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B2))
             => ( aa_real_real(F,X2) = aa_real_real(F,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10197_card__num1,axiom,
    finite6454714172617411597l_num1(top_to3689904429138878997l_num1) = one_one_nat ).

% card_num1
tff(fact_10198_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ahe,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10199_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_uq,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10200_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_ahz,Uu))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10201_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tu,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10202_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ajt,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10203_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_akn,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10204_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ajv,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_10205_ATP_Olambda__8,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_wt,Uu) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),one_one_complex) ).

% ATP.lambda_8
tff(fact_10206_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_wv,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_9
tff(fact_10207_ATP_Olambda__10,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_wu,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_10
tff(fact_10208_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_wx,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_11
tff(fact_10209_ATP_Olambda__12,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ww,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_12
tff(fact_10210_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ajx,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_13
tff(fact_10211_ATP_Olambda__14,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_aiz,Uu))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uu)) ) ).

% ATP.lambda_14
tff(fact_10212_ATP_Olambda__15,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ajn,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ahg,Uu)) ).

% ATP.lambda_15
tff(fact_10213_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ajy,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_16
tff(fact_10214_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aii,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_17
tff(fact_10215_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aih,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_18
tff(fact_10216_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ahr,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_19
tff(fact_10217_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_afv(Uu),Uua) = aa_Cod1783974652533396728nteger(if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_z3403309356797280102nteger) ).

% ATP.lambda_20
tff(fact_10218_ATP_Olambda__21,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_afw(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_21
tff(fact_10219_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_afz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_22
tff(fact_10220_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_afx(Uu),Uua) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_rat) ).

% ATP.lambda_23
tff(fact_10221_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_afy,Uu),Uua) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_24
tff(fact_10222_ATP_Olambda__25,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uo(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),zero_zero_real),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_25
tff(fact_10223_ATP_Olambda__26,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aif,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_26
tff(fact_10224_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_afq(Uu),Uua) = aa_Cod1783974652533396728nteger(if_Code_integer(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_z3403309356797280102nteger) ).

% ATP.lambda_27
tff(fact_10225_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_afr(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_28
tff(fact_10226_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_afu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_29
tff(fact_10227_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_afs(Uu),Uua) = aa_rat_rat(if_rat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_rat) ).

% ATP.lambda_30
tff(fact_10228_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_aft,Uu),Uua) = aa_int_int(aa_int_fun_int_int(if_int(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_31
tff(fact_10229_ATP_Olambda__32,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_32
tff(fact_10230_ATP_Olambda__33,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aha,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_33
tff(fact_10231_ATP_Olambda__34,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_34
tff(fact_10232_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_aen(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_35
tff(fact_10233_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_aep,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_36
tff(fact_10234_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_aeo(Uu),Uua) = aa_rat_rat(divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_37
tff(fact_10235_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aia,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_38
tff(fact_10236_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ahv,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_39
tff(fact_10237_ATP_Olambda__40,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aie(Uu),Uua))
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).

% ATP.lambda_40
tff(fact_10238_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_aib,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_41
tff(fact_10239_ATP_Olambda__42,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_aic(Uu),Uua))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_42
tff(fact_10240_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_43
tff(fact_10241_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_44
tff(fact_10242_ATP_Olambda__45,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aju(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_45
tff(fact_10243_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ade,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_46
tff(fact_10244_ATP_Olambda__47,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_add(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_47
tff(fact_10245_ATP_Olambda__48,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_wy,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,Uu),Uua)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_48
tff(fact_10246_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xa,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_49
tff(fact_10247_ATP_Olambda__50,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wz(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_50
tff(fact_10248_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_agr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_51
tff(fact_10249_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_52
tff(fact_10250_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ako,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_53
tff(fact_10251_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_agc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_54
tff(fact_10252_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_agb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_55
tff(fact_10253_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_gk,Uu),Uua))
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_56
tff(fact_10254_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_gj,Uu),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_57
tff(fact_10255_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ake,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_58
tff(fact_10256_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_akl,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_59
tff(fact_10257_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_akp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_60
tff(fact_10258_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_61
tff(fact_10259_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ahy,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_62
tff(fact_10260_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_acj(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_63
tff(fact_10261_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ack(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_64
tff(fact_10262_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_acm(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_65
tff(fact_10263_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_acl(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_66
tff(fact_10264_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rf(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_67
tff(fact_10265_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rc(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_68
tff(fact_10266_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_re(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_69
tff(fact_10267_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rd(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_70
tff(fact_10268_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qo(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_71
tff(fact_10269_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qm(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_72
tff(fact_10270_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qn(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_73
tff(fact_10271_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_abn(Uu),Uua) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_74
tff(fact_10272_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_abl(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_75
tff(fact_10273_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qt(Uu),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_76
tff(fact_10274_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qr(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_77
tff(fact_10275_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_le(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uua)) ).

% ATP.lambda_78
tff(fact_10276_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lg(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_79
tff(fact_10277_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lf(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_80
tff(fact_10278_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rb(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_81
tff(fact_10279_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qz(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_82
tff(fact_10280_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ra(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_83
tff(fact_10281_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qv(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_84
tff(fact_10282_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qy(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_85
tff(fact_10283_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qw(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_86
tff(fact_10284_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_87
tff(fact_10285_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sm(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_88
tff(fact_10286_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sk(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_89
tff(fact_10287_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sl(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_90
tff(fact_10288_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_adc(Uu),Uua) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_91
tff(fact_10289_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_adb(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_92
tff(fact_10290_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ada(Uu),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_93
tff(fact_10291_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_acz(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_94
tff(fact_10292_ATP_Olambda__95,axiom,
    ! [Uu: fun_Co1900716531650368264omplex,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_pg(Uu),Uua))
    <=> ( aa_Cod5621350732091595593omplex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_95
tff(fact_10293_ATP_Olambda__96,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_pi(Uu),Uua))
    <=> ( aa_Code_integer_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_96
tff(fact_10294_ATP_Olambda__97,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_pl(Uu),Uua))
    <=> ( aa_Code_integer_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_97
tff(fact_10295_ATP_Olambda__98,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_po(Uu),Uua))
    <=> ( aa_Code_integer_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_98
tff(fact_10296_ATP_Olambda__99,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_pj(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_99
tff(fact_10297_ATP_Olambda__100,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_pm(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_100
tff(fact_10298_ATP_Olambda__101,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_pf(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_101
tff(fact_10299_ATP_Olambda__102,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ph(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_102
tff(fact_10300_ATP_Olambda__103,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_pk(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_103
tff(fact_10301_ATP_Olambda__104,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_pn(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_104
tff(fact_10302_ATP_Olambda__105,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_aaz(Uu),Uua))
    <=> ( aa_Code_integer_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_105
tff(fact_10303_ATP_Olambda__106,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_abc(Uu),Uua))
    <=> ( aa_Code_integer_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_106
tff(fact_10304_ATP_Olambda__107,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_abf(Uu),Uua))
    <=> ( aa_Code_integer_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_107
tff(fact_10305_ATP_Olambda__108,axiom,
    ! [Uu: fun_complex_assn,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_aba(Uu),Uua))
    <=> ( aa_complex_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_108
tff(fact_10306_ATP_Olambda__109,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_abd(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_109
tff(fact_10307_ATP_Olambda__110,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_abg(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_110
tff(fact_10308_ATP_Olambda__111,axiom,
    ! [Uu: fun_int_assn,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_aay(Uu),Uua))
    <=> ( aa_int_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_111
tff(fact_10309_ATP_Olambda__112,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_abb(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_112
tff(fact_10310_ATP_Olambda__113,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_abe(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_113
tff(fact_10311_ATP_Olambda__114,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_abh(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_114
tff(fact_10312_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_akk(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_aju(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_115
tff(fact_10313_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_akj(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_aju(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_116
tff(fact_10314_ATP_Olambda__117,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tw,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_117
tff(fact_10315_ATP_Olambda__118,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_118
tff(fact_10316_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aid,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_119
tff(fact_10317_ATP_Olambda__120,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_120
tff(fact_10318_ATP_Olambda__121,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ahc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_121
tff(fact_10319_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_act,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_122
tff(fact_10320_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_acs(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_123
tff(fact_10321_ATP_Olambda__124,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ahw,Uu),Uua))
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_124
tff(fact_10322_ATP_Olambda__125,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_wm,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_125
tff(fact_10323_ATP_Olambda__126,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_126
tff(fact_10324_ATP_Olambda__127,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wn(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_127
tff(fact_10325_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ags,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_128
tff(fact_10326_ATP_Olambda__129,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_akf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_129
tff(fact_10327_ATP_Olambda__130,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ahs,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).

% ATP.lambda_130
tff(fact_10328_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_akd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_131
tff(fact_10329_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aka,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_132
tff(fact_10330_ATP_Olambda__133,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ala(Uu),Uua) = groups6591440286371151544t_real(Uu,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_133
tff(fact_10331_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_134
tff(fact_10332_ATP_Olambda__135,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_acb(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_135
tff(fact_10333_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ace,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_136
tff(fact_10334_ATP_Olambda__137,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_acd,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_137
tff(fact_10335_ATP_Olambda__138,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(aTP_Lamm_go(Uu),Uua))
    <=> pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(Uua),Uu)) ) ).

% ATP.lambda_138
tff(fact_10336_ATP_Olambda__139,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_gp(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_139
tff(fact_10337_ATP_Olambda__140,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_gn(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Uua),Uu)) ) ).

% ATP.lambda_140
tff(fact_10338_ATP_Olambda__141,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_gq(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_10339_ATP_Olambda__142,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_acu,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_10340_ATP_Olambda__143,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_acv,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_10341_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gl,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_144
tff(fact_10342_ATP_Olambda__145,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_acw,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_145
tff(fact_10343_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ain,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_146
tff(fact_10344_ATP_Olambda__147,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_sf,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_147
tff(fact_10345_ATP_Olambda__148,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_sc,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_148
tff(fact_10346_ATP_Olambda__149,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_sd,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_10347_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gm,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_10348_ATP_Olambda__151,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_se,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_151
tff(fact_10349_ATP_Olambda__152,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_am,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_152
tff(fact_10350_ATP_Olambda__153,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_an(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_153
tff(fact_10351_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_154
tff(fact_10352_ATP_Olambda__155,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ap,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_155
tff(fact_10353_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_156
tff(fact_10354_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ajb,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_157
tff(fact_10355_ATP_Olambda__158,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_aip(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu) ).

% ATP.lambda_158
tff(fact_10356_ATP_Olambda__159,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aio,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_159
tff(fact_10357_ATP_Olambda__160,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ajd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_160
tff(fact_10358_ATP_Olambda__161,axiom,
    ! [Uu: complex,Uua: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_dt,Uu),Uua))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,Uua),Uu)) ) ).

% ATP.lambda_161
tff(fact_10359_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_du,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_162
tff(fact_10360_ATP_Olambda__163,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dv,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_163
tff(fact_10361_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aga,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_164
tff(fact_10362_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajz,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_165
tff(fact_10363_ATP_Olambda__166,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uua),Uu)) ) ).

% ATP.lambda_166
tff(fact_10364_ATP_Olambda__167,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ae(Uu),Uua))
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_167
tff(fact_10365_ATP_Olambda__168,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ad(Uu),Uua))
    <=> pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).

% ATP.lambda_168
tff(fact_10366_ATP_Olambda__169,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_169
tff(fact_10367_ATP_Olambda__170,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_170
tff(fact_10368_ATP_Olambda__171,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ac(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_171
tff(fact_10369_ATP_Olambda__172,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_bx(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_172
tff(fact_10370_ATP_Olambda__173,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_bz(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_173
tff(fact_10371_ATP_Olambda__174,axiom,
    ! [Uu: complex,Uua: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_by,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_174
tff(fact_10372_ATP_Olambda__175,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ca,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_175
tff(fact_10373_ATP_Olambda__176,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cb,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_176
tff(fact_10374_ATP_Olambda__177,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_hw(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua))) ) ).

% ATP.lambda_177
tff(fact_10375_ATP_Olambda__178,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_hq(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Pro6801748904974264397nt_nat(Uu,Uua))) ) ).

% ATP.lambda_178
tff(fact_10376_ATP_Olambda__179,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_hy(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu,Uua))) ) ).

% ATP.lambda_179
tff(fact_10377_ATP_Olambda__180,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_hs(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua))) ) ).

% ATP.lambda_180
tff(fact_10378_ATP_Olambda__181,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_hu(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua))) ) ).

% ATP.lambda_181
tff(fact_10379_ATP_Olambda__182,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tt(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_182
tff(fact_10380_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ts(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_183
tff(fact_10381_ATP_Olambda__184,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aky(Uu),Uua))
    <=> pp(aa_real_bool(Uu,aa_real_real(inverse_inverse_real,Uua))) ) ).

% ATP.lambda_184
tff(fact_10382_ATP_Olambda__185,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_aja(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_185
tff(fact_10383_ATP_Olambda__186,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_akw(Uu),Uua))
    <=> pp(aa_real_bool(Uu,aa_real_real(uminus_uminus_real,Uua))) ) ).

% ATP.lambda_186
tff(fact_10384_ATP_Olambda__187,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_vy(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_187
tff(fact_10385_ATP_Olambda__188,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pq(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_188
tff(fact_10386_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_akt(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_189
tff(fact_10387_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qk(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_190
tff(fact_10388_ATP_Olambda__191,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pp(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_191
tff(fact_10389_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ql(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_192
tff(fact_10390_ATP_Olambda__193,axiom,
    ! [Uu: fun_li7023630697154767927t_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_ex(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_nat6700791848136694008i_assn(aTP_Lamm_ew(Uu),Uua)) ).

% ATP.lambda_193
tff(fact_10391_ATP_Olambda__194,axiom,
    ! [Uu: fun_li3309467822378783623t_assn,Uua: option_nat] : aa_option_nat_assn(aTP_Lamm_ev(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_opt2780285371362742984i_assn(aTP_Lamm_eu(Uu),Uua)) ).

% ATP.lambda_194
tff(fact_10392_ATP_Olambda__195,axiom,
    ! [Uu: fun_li4779488348070916006l_assn,Uua: bool] : aa_bool_assn(aTP_Lamm_et(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_boo8579855676982511045i_assn(aTP_Lamm_es(Uu),Uua)) ).

% ATP.lambda_195
tff(fact_10393_ATP_Olambda__196,axiom,
    ! [Uu: fun_li8825114373765265308i_assn,Uua: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aTP_Lamm_er(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_VEB3739350836417479773i_assn(aTP_Lamm_eq(Uu),Uua)) ).

% ATP.lambda_196
tff(fact_10394_ATP_Olambda__197,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_ajr(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ajq(Uu),Uua)) ).

% ATP.lambda_197
tff(fact_10395_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ahl(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_198
tff(fact_10396_ATP_Olambda__199,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_akq(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_199
tff(fact_10397_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ahm,Uu),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_200
tff(fact_10398_ATP_Olambda__201,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_agw(Uu),Uua) = aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_201
tff(fact_10399_ATP_Olambda__202,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_agy(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_202
tff(fact_10400_ATP_Olambda__203,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_agx(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_203
tff(fact_10401_ATP_Olambda__204,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_wq(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_204
tff(fact_10402_ATP_Olambda__205,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_wp(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_205
tff(fact_10403_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ws(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_206
tff(fact_10404_ATP_Olambda__207,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_wr(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_207
tff(fact_10405_ATP_Olambda__208,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_akc,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_208
tff(fact_10406_ATP_Olambda__209,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_om(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_10407_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_ol(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_10408_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_on(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_10409_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ajw(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_10410_ATP_Olambda__213,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_do(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_10411_ATP_Olambda__214,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_ka(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_10412_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ke(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_10413_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kd(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_10414_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_kc(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_10415_ATP_Olambda__218,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_kb(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_10416_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_ahj(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_10417_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_fun_assn_assn(aTP_Lamm_zz(Uu),Uua) = times_times_assn(aa_nat_assn(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_10418_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_aaa(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_10419_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_aab(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_10420_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_aad(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_10421_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_aac(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_10422_ATP_Olambda__225,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_kf(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_10423_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_kg(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10424_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_xe(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_10425_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_xf(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_10426_ATP_Olambda__229,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_kh(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_10427_ATP_Olambda__230,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_ki(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_10428_ATP_Olambda__231,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_kj(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_10429_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_acn(Uu),Uua) = aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_10430_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_acr(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_10431_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_aco(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_10432_ATP_Olambda__235,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_acq(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_235
tff(fact_10433_ATP_Olambda__236,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_acp(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_236
tff(fact_10434_ATP_Olambda__237,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_alf(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_237
tff(fact_10435_ATP_Olambda__238,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_alb(Uu),Uua) = aa_real_real(arsinh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_238
tff(fact_10436_ATP_Olambda__239,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ale(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_239
tff(fact_10437_ATP_Olambda__240,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ahi(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_240
tff(fact_10438_ATP_Olambda__241,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aki(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_241
tff(fact_10439_ATP_Olambda__242,axiom,
    ! [Uu: fun_VE3708348504534315036i_assn,Uua: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aTP_Lamm_ej(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_VEB3739350836417479773i_assn(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_10440_ATP_Olambda__243,axiom,
    ! [Uu: fun_op8552747401449489031i_assn,Uua: option_nat] : aa_option_nat_assn(aTP_Lamm_en(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_opt2780285371362742984i_assn(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_10441_ATP_Olambda__244,axiom,
    ! [Uu: fun_bo5386306568165246356i_assn,Uua: bool] : aa_bool_assn(aTP_Lamm_el(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_boo8579855676982511045i_assn(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_10442_ATP_Olambda__245,axiom,
    ! [Uu: fun_na8005944528675429559i_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_ep(Uu),Uua) = ex_ass463751140784270563_VEBTi(aa_nat6700791848136694008i_assn(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_10443_ATP_Olambda__246,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_ajs(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_10444_ATP_Olambda__247,axiom,
    ! [Uu: fun_Co8822979462182099027r_bool,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_uh(Uu),Uua))
    <=> ~ pp(aa_Code_integer_bool(Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_10445_ATP_Olambda__248,axiom,
    ! [Uu: fun_Pr1572506220757514509t_bool,Uua: product_unit] :
      ( pp(aa_Product_unit_bool(aTP_Lamm_uj(Uu),Uua))
    <=> ~ pp(aa_Product_unit_bool(Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_10446_ATP_Olambda__249,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_uf(Uu),Uua))
    <=> ~ pp(aa_Pro4501881432599481050t_bool(Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_10447_ATP_Olambda__250,axiom,
    ! [Uu: fun_Nu28140169051909368851_bool,Uua: numeral_num1] :
      ( pp(aa_Numeral_num1_bool(aTP_Lamm_uk(Uu),Uua))
    <=> ~ pp(aa_Numeral_num1_bool(Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_10448_ATP_Olambda__251,axiom,
    ! [Uu: fun_complex_bool,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ui(Uu),Uua))
    <=> ~ pp(aa_complex_bool(Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_10449_ATP_Olambda__252,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ul(Uu),Uua))
    <=> ~ pp(aa_real_bool(Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_10450_ATP_Olambda__253,axiom,
    ! [Uu: fun_bool_bool,Uua: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_um(Uu),Uua))
    <=> ~ pp(aa_bool_bool(Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_10451_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_un(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_10452_ATP_Olambda__255,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ug(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_10453_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_up(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_256
tff(fact_10454_ATP_Olambda__257,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_257
tff(fact_10455_ATP_Olambda__258,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ahq(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_258
tff(fact_10456_ATP_Olambda__259,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_aho(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_259
tff(fact_10457_ATP_Olambda__260,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ahp(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_260
tff(fact_10458_ATP_Olambda__261,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_wf(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_261
tff(fact_10459_ATP_Olambda__262,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_wi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_262
tff(fact_10460_ATP_Olambda__263,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_263
tff(fact_10461_ATP_Olambda__264,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_wg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_264
tff(fact_10462_ATP_Olambda__265,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_kk(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu),Uub)),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_265
tff(fact_10463_ATP_Olambda__266,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_kp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_266
tff(fact_10464_ATP_Olambda__267,axiom,
    ! [Uu: code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ko(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uu),Uub)),aa_Cod5621350732091595593omplex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_267
tff(fact_10465_ATP_Olambda__268,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_ks(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uu),Uub)),aa_Code_integer_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_268
tff(fact_10466_ATP_Olambda__269,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_kt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_269
tff(fact_10467_ATP_Olambda__270,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_kl(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_270
tff(fact_10468_ATP_Olambda__271,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_271
tff(fact_10469_ATP_Olambda__272,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_km(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_272
tff(fact_10470_ATP_Olambda__273,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_kn(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_273
tff(fact_10471_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_kr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_274
tff(fact_10472_ATP_Olambda__275,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] : aa_VEBT_VEBT_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_xq(Uu),Uua),Uub) = if_assn(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu),Uub),aa_VEBT_VEBT_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_275
tff(fact_10473_ATP_Olambda__276,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_xw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_276
tff(fact_10474_ATP_Olambda__277,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] : aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xu(Uu),Uua),Uub) = if_assn(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uu),Uub),aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_277
tff(fact_10475_ATP_Olambda__278,axiom,
    ! [Uu: complex,Uua: fun_complex_assn,Uub: complex] : aa_complex_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_xv(Uu),Uua),Uub) = if_assn(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_278
tff(fact_10476_ATP_Olambda__279,axiom,
    ! [Uu: real,Uua: fun_real_assn,Uub: real] : aa_real_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_xr(Uu),Uua),Uub) = if_assn(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_279
tff(fact_10477_ATP_Olambda__280,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_xx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_280
tff(fact_10478_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] : aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xs(Uu),Uua),Uub) = if_assn(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_281
tff(fact_10479_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_282
tff(fact_10480_ATP_Olambda__283,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] : aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xt(Uu),Uua),Uub) = if_assn(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_283
tff(fact_10481_ATP_Olambda__284,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_xz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_284
tff(fact_10482_ATP_Olambda__285,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_ku(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu)),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_285
tff(fact_10483_ATP_Olambda__286,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_kz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_286
tff(fact_10484_ATP_Olambda__287,axiom,
    ! [Uu: code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ky(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uu)),aa_Cod5621350732091595593omplex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_287
tff(fact_10485_ATP_Olambda__288,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_lc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uu)),aa_Code_integer_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_288
tff(fact_10486_ATP_Olambda__289,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ld(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_289
tff(fact_10487_ATP_Olambda__290,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_kv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_290
tff(fact_10488_ATP_Olambda__291,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_la(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_291
tff(fact_10489_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kw(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_292
tff(fact_10490_ATP_Olambda__293,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_293
tff(fact_10491_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ve(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_294
tff(fact_10492_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_vf(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_295
tff(fact_10493_ATP_Olambda__296,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_kx(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_296
tff(fact_10494_ATP_Olambda__297,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_lb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_297
tff(fact_10495_ATP_Olambda__298,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] : aa_VEBT_VEBT_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_xg(Uu),Uua),Uub) = if_assn(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu),aa_VEBT_VEBT_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_298
tff(fact_10496_ATP_Olambda__299,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_xm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu)),aa_VEBT_VEBT_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_299
tff(fact_10497_ATP_Olambda__300,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] : aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xk(Uu),Uua),Uub) = if_assn(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uu),aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_300
tff(fact_10498_ATP_Olambda__301,axiom,
    ! [Uu: complex,Uua: fun_complex_assn,Uub: complex] : aa_complex_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_xl(Uu),Uua),Uub) = if_assn(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_301
tff(fact_10499_ATP_Olambda__302,axiom,
    ! [Uu: real,Uua: fun_real_assn,Uub: real] : aa_real_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_xh(Uu),Uua),Uub) = if_assn(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_302
tff(fact_10500_ATP_Olambda__303,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_xn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_303
tff(fact_10501_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] : aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xi(Uu),Uua),Uub) = if_assn(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_304
tff(fact_10502_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_305
tff(fact_10503_ATP_Olambda__306,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] : aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xj(Uu),Uua),Uub) = if_assn(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_306
tff(fact_10504_ATP_Olambda__307,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_xp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_307
tff(fact_10505_ATP_Olambda__308,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_wb(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(Uu,Uub)),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_308
tff(fact_10506_ATP_Olambda__309,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_we(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uu,Uub)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_309
tff(fact_10507_ATP_Olambda__310,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uu,Uub)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_310
tff(fact_10508_ATP_Olambda__311,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_wc(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(Uu,Uub)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_311
tff(fact_10509_ATP_Olambda__312,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_ou(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_312
tff(fact_10510_ATP_Olambda__313,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_pa(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_313
tff(fact_10511_ATP_Olambda__314,axiom,
    ! [Uu: fun_Co1900716531650368264omplex,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun5444226037728322975omplex(aTP_Lamm_oz(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_Code_integer_bool(Uua,Uub)),aa_Cod5621350732091595593omplex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_314
tff(fact_10512_ATP_Olambda__315,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] : aa_Code_integer_real(aa_fun1813952354503624093r_real(aTP_Lamm_pd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Code_integer_bool(Uua,Uub)),aa_Code_integer_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_315
tff(fact_10513_ATP_Olambda__316,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_pe(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_316
tff(fact_10514_ATP_Olambda__317,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_ov(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_real_bool(Uua,Uub)),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_317
tff(fact_10515_ATP_Olambda__318,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_pb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_318
tff(fact_10516_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_ow(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(Uua,Uub)),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_319
tff(fact_10517_ATP_Olambda__320,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_ox(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_int_bool(Uua,Uub)),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_320
tff(fact_10518_ATP_Olambda__321,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_pc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_321
tff(fact_10519_ATP_Olambda__322,axiom,
    ! [Uu: fun_VEBT_VEBT_assn,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_assn(aa_fun158440353102879243T_assn(aTP_Lamm_aae(Uu),Uua),Uub) = if_assn(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_322
tff(fact_10520_ATP_Olambda__323,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_aak(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_323
tff(fact_10521_ATP_Olambda__324,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] : aa_Code_integer_assn(aa_fun8033543692415806745r_assn(aTP_Lamm_aai(Uu),Uua),Uub) = if_assn(aa_Code_integer_bool(Uua,Uub),aa_Code_integer_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_324
tff(fact_10522_ATP_Olambda__325,axiom,
    ! [Uu: fun_complex_assn,Uua: fun_complex_bool,Uub: complex] : aa_complex_assn(aa_fun3623554829249773913x_assn(aTP_Lamm_aaj(Uu),Uua),Uub) = if_assn(aa_complex_bool(Uua,Uub),aa_complex_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_325
tff(fact_10523_ATP_Olambda__326,axiom,
    ! [Uu: fun_real_assn,Uua: fun_real_bool,Uub: real] : aa_real_assn(aa_fun179583594588223833l_assn(aTP_Lamm_aaf(Uu),Uua),Uub) = if_assn(aa_real_bool(Uua,Uub),aa_real_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_326
tff(fact_10524_ATP_Olambda__327,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_aal(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_327
tff(fact_10525_ATP_Olambda__328,axiom,
    ! [Uu: fun_nat_assn,Uua: fun_nat_bool,Uub: nat] : aa_nat_assn(aa_fun1954384836919635929t_assn(aTP_Lamm_aag(Uu),Uua),Uub) = if_assn(aa_nat_bool(Uua,Uub),aa_nat_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_328
tff(fact_10526_ATP_Olambda__329,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_bool,Uub: nat] : aa_nat_real(aa_fun1750223370215434333t_real(aTP_Lamm_aam(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uua,Uub)),aa_nat_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_329
tff(fact_10527_ATP_Olambda__330,axiom,
    ! [Uu: fun_int_assn,Uua: fun_int_bool,Uub: int] : aa_int_assn(aa_fun6196142345855524569t_assn(aTP_Lamm_aah(Uu),Uua),Uub) = if_assn(aa_int_bool(Uua,Uub),aa_int_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_330
tff(fact_10528_ATP_Olambda__331,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_aan(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_331
tff(fact_10529_ATP_Olambda__332,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_akr(Uu),Uua),Uub))
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_332
tff(fact_10530_ATP_Olambda__333,axiom,
    ! [Uu: fun_VE3708348504534315036i_assn,Uua: list_VEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_lis5319508155010942429i_assn(aTP_Lamm_ei(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_VEB3739350836417479773i_assn(Uu,Uub),Uua) ).

% ATP.lambda_333
tff(fact_10531_ATP_Olambda__334,axiom,
    ! [Uu: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_mj(Uu),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu,Uub),Uua) ).

% ATP.lambda_334
tff(fact_10532_ATP_Olambda__335,axiom,
    ! [Uu: fun_VE2989161857680785728t_real,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_nat4323386229672267365T_real(aTP_Lamm_vj(Uu),Uua),Uub) = aa_nat_real(aa_VEB3736018289492012273t_real(Uu,Uub),Uua) ).

% ATP.lambda_335
tff(fact_10533_ATP_Olambda__336,axiom,
    ! [Uu: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_lm(Uu),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_10534_ATP_Olambda__337,axiom,
    ! [Uu: fun_VE7331017413731640256at_int,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_yr(Uu),Uua),Uub) = aa_nat_int(aa_VEB6167613734120275313at_int(Uu,Uub),Uua) ).

% ATP.lambda_337
tff(fact_10535_ATP_Olambda__338,axiom,
    ! [Uu: fun_VE294566951569886236nt_int,Uua: int,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_int1343382402272391561BT_int(aTP_Lamm_ze(Uu),Uua),Uub) = aa_int_int(aa_VEB7166967446021329869nt_int(Uu,Uub),Uua) ).

% ATP.lambda_338
tff(fact_10536_ATP_Olambda__339,axiom,
    ! [Uu: fun_Co8905303033605938227omplex,Uua: complex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_com7552888802784664052omplex(aTP_Lamm_nb(Uu),Uua),Uub) = aa_complex_complex(aa_Cod3397930797123873268omplex(Uu,Uub),Uua) ).

% ATP.lambda_339
tff(fact_10537_ATP_Olambda__340,axiom,
    ! [Uu: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_lz(Uu),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu,Uub),Uua) ).

% ATP.lambda_340
tff(fact_10538_ATP_Olambda__341,axiom,
    ! [Uu: fun_Co822469698833053651at_int,Uua: nat,Uub: code_integer] : aa_Code_integer_int(aa_nat1674686023564409748er_int(aTP_Lamm_yx(Uu),Uua),Uub) = aa_nat_int(aa_Cod4492442032739075476at_int(Uu,Uub),Uua) ).

% ATP.lambda_341
tff(fact_10539_ATP_Olambda__342,axiom,
    ! [Uu: fun_Co3009391273526075439nt_int,Uua: int,Uub: code_integer] : aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_zm(Uu),Uua),Uub) = aa_int_int(aa_Cod5491795744640130032nt_int(Uu,Uub),Uua) ).

% ATP.lambda_342
tff(fact_10540_ATP_Olambda__343,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ob(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_343
tff(fact_10541_ATP_Olambda__344,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: nat,Uub: complex] : aa_complex_complex(aa_nat3829984526574172007omplex(aTP_Lamm_vq(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(Uu,Uub),Uua) ).

% ATP.lambda_344
tff(fact_10542_ATP_Olambda__345,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_me(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_345
tff(fact_10543_ATP_Olambda__346,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_za(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_346
tff(fact_10544_ATP_Olambda__347,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_zr(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_347
tff(fact_10545_ATP_Olambda__348,axiom,
    ! [Uu: fun_op8552747401449489031i_assn,Uua: list_VEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aTP_Lamm_em(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_opt2780285371362742984i_assn(Uu,Uub),Uua) ).

% ATP.lambda_348
tff(fact_10546_ATP_Olambda__349,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_mn(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_349
tff(fact_10547_ATP_Olambda__350,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_vl(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_350
tff(fact_10548_ATP_Olambda__351,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_lq(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_351
tff(fact_10549_ATP_Olambda__352,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_yu(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_352
tff(fact_10550_ATP_Olambda__353,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_zi(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_353
tff(fact_10551_ATP_Olambda__354,axiom,
    ! [Uu: fun_li8825114373765265308i_assn,Uua: vEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_VEB3739350836417479773i_assn(aTP_Lamm_eq(Uu),Uua),Uub) = aa_VEBT_VEBTi_assn(aa_lis5319508155010942429i_assn(Uu,Uub),Uua) ).

% ATP.lambda_354
tff(fact_10552_ATP_Olambda__355,axiom,
    ! [Uu: fun_li3309467822378783623t_assn,Uua: option_nat,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_opt2780285371362742984i_assn(aTP_Lamm_eu(Uu),Uua),Uub) = aa_option_nat_assn(aa_lis7387135221810377032t_assn(Uu,Uub),Uua) ).

% ATP.lambda_355
tff(fact_10553_ATP_Olambda__356,axiom,
    ! [Uu: fun_li4779488348070916006l_assn,Uua: bool,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_boo8579855676982511045i_assn(aTP_Lamm_es(Uu),Uua),Uub) = aa_bool_assn(aa_lis822886067112437519l_assn(Uu,Uub),Uua) ).

% ATP.lambda_356
tff(fact_10554_ATP_Olambda__357,axiom,
    ! [Uu: fun_li7023630697154767927t_assn,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aTP_Lamm_ew(Uu),Uua),Uub) = aa_nat_assn(aa_lis7348694692495225848t_assn(Uu,Uub),Uua) ).

% ATP.lambda_357
tff(fact_10555_ATP_Olambda__358,axiom,
    ! [Uu: fun_bo5386306568165246356i_assn,Uua: list_VEBT_VEBTi,Uub: bool] : aa_bool_assn(aa_lis822886067112437519l_assn(aTP_Lamm_ek(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_boo8579855676982511045i_assn(Uu,Uub),Uua) ).

% ATP.lambda_358
tff(fact_10556_ATP_Olambda__359,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mr(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_359
tff(fact_10557_ATP_Olambda__360,axiom,
    ! [Uu: fun_na8005944528675429559i_assn,Uua: list_VEBT_VEBTi,Uub: nat] : aa_nat_assn(aa_lis7348694692495225848t_assn(aTP_Lamm_eo(Uu),Uua),Uub) = aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(Uu,Uub),Uua) ).

% ATP.lambda_360
tff(fact_10558_ATP_Olambda__361,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_oe(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_361
tff(fact_10559_ATP_Olambda__362,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_362
tff(fact_10560_ATP_Olambda__363,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_yb(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_363
tff(fact_10561_ATP_Olambda__364,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ye(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_364
tff(fact_10562_ATP_Olambda__365,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_mw(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_365
tff(fact_10563_ATP_Olambda__366,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_vn(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_366
tff(fact_10564_ATP_Olambda__367,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_lu(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_367
tff(fact_10565_ATP_Olambda__368,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_vt(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_368
tff(fact_10566_ATP_Olambda__369,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_oh(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_369
tff(fact_10567_ATP_Olambda__370,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_wj(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_complex),zero_zero_complex)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_370
tff(fact_10568_ATP_Olambda__371,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_real),zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_371
tff(fact_10569_ATP_Olambda__372,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_wl(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_int),zero_zero_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_372
tff(fact_10570_ATP_Olambda__373,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_abm(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_yb(Uu),Uub),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_373
tff(fact_10571_ATP_Olambda__374,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abo(Uu),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uub),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_374
tff(fact_10572_ATP_Olambda__375,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qu(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oe(Uu),Uub),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_375
tff(fact_10573_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qs(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_376
tff(fact_10574_ATP_Olambda__377,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aiy(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aix(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_377
tff(fact_10575_ATP_Olambda__378,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aiw(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aiv(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_378
tff(fact_10576_ATP_Olambda__379,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aiu(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ait(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_379
tff(fact_10577_ATP_Olambda__380,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ais(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_air(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_380
tff(fact_10578_ATP_Olambda__381,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_aim(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_ail(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_381
tff(fact_10579_ATP_Olambda__382,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_aik(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_aij(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_382
tff(fact_10580_ATP_Olambda__383,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_yf(Uu),Uua),Uub) = groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ye(Uu),Uub),Uua) ).

% ATP.lambda_383
tff(fact_10581_ATP_Olambda__384,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_yc(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_yb(Uu),Uub),Uua) ).

% ATP.lambda_384
tff(fact_10582_ATP_Olambda__385,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_yh(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_vt(Uu),Uub),Uua) ).

% ATP.lambda_385
tff(fact_10583_ATP_Olambda__386,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_yj(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_oh(Uu),Uub),Uua) ).

% ATP.lambda_386
tff(fact_10584_ATP_Olambda__387,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_yl(Uu),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uub),Uua) ).

% ATP.lambda_387
tff(fact_10585_ATP_Olambda__388,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: nat] : aa_nat_int(aa_fun8710705519803144921at_int(aTP_Lamm_vu(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_vt(Uua),Uub),Uu) ).

% ATP.lambda_388
tff(fact_10586_ATP_Olambda__389,axiom,
    ! [Uu: set_complex,Uua: fun_co3258106519541737638omplex,Uub: nat] : aa_nat_complex(aa_fun5003373910569599327omplex(aTP_Lamm_vr(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(aTP_Lamm_vq(Uua),Uub),Uu) ).

% ATP.lambda_389
tff(fact_10587_ATP_Olambda__390,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun2842512154077127769t_real(aTP_Lamm_vo(Uu),Uua),Uub) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_vn(Uua),Uub),Uu) ).

% ATP.lambda_390
tff(fact_10588_ATP_Olambda__391,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: nat] : aa_nat_real(aa_fun628238740774635481t_real(aTP_Lamm_vm(Uu),Uua),Uub) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_vl(Uua),Uub),Uu) ).

% ATP.lambda_391
tff(fact_10589_ATP_Olambda__392,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2989161857680785728t_real,Uub: nat] : aa_nat_real(aa_fun6147343916763992189t_real(aTP_Lamm_vk(Uu),Uua),Uub) = groups2240296850493347238T_real(aa_nat4323386229672267365T_real(aTP_Lamm_vj(Uua),Uub),Uu) ).

% ATP.lambda_392
tff(fact_10590_ATP_Olambda__393,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_oi(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_oh(Uu),Uub),Uua) ).

% ATP.lambda_393
tff(fact_10591_ATP_Olambda__394,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_of(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oe(Uu),Uub),Uua) ).

% ATP.lambda_394
tff(fact_10592_ATP_Olambda__395,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_vs(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oe(Uua),Uub),Uu) ).

% ATP.lambda_395
tff(fact_10593_ATP_Olambda__396,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_oc(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ob(Uu),Uub),Uua) ).

% ATP.lambda_396
tff(fact_10594_ATP_Olambda__397,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_nz(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uub)),Uua) ).

% ATP.lambda_397
tff(fact_10595_ATP_Olambda__398,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_vp(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uua),Uub)),Uu) ).

% ATP.lambda_398
tff(fact_10596_ATP_Olambda__399,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_399
tff(fact_10597_ATP_Olambda__400,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ajh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_400
tff(fact_10598_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_adg(Uu),Uua),Uub))
    <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acy(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_401
tff(fact_10599_ATP_Olambda__402,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_adf(Uu),Uua),Uub))
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_acx(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_402
tff(fact_10600_ATP_Olambda__403,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lh(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_403
tff(fact_10601_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_404
tff(fact_10602_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_li(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_405
tff(fact_10603_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_406
tff(fact_10604_ATP_Olambda__407,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_agz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_407
tff(fact_10605_ATP_Olambda__408,axiom,
    ! [Uu: set_VEBT_VEBTi,Uua: nat,Uub: list_VEBT_VEBTi] :
      ( pp(aa_lis8521252006704567508i_bool(aa_nat5293450157895600989i_bool(aTP_Lamm_ga(Uu),Uua),Uub))
    <=> ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Uub)),Uua)) ) ) ).

% ATP.lambda_408
tff(fact_10606_ATP_Olambda__409,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_fw(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).

% ATP.lambda_409
tff(fact_10607_ATP_Olambda__410,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( pp(aa_lis9036377640098331692r_bool(aa_nat3680331675976215733r_bool(aTP_Lamm_fu(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua)) ) ) ).

% ATP.lambda_410
tff(fact_10608_ATP_Olambda__411,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_fv(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_411
tff(fact_10609_ATP_Olambda__412,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_fx(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_real(Uub)),Uua)) ) ) ).

% ATP.lambda_412
tff(fact_10610_ATP_Olambda__413,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fy(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).

% ATP.lambda_413
tff(fact_10611_ATP_Olambda__414,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fz(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_414
tff(fact_10612_ATP_Olambda__415,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_gb(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).

% ATP.lambda_415
tff(fact_10613_ATP_Olambda__416,axiom,
    ! [Uu: set_VEBT_VEBTi,Uua: nat,Uub: list_VEBT_VEBTi] :
      ( pp(aa_lis8521252006704567508i_bool(aa_nat5293450157895600989i_bool(aTP_Lamm_fs(Uu),Uua),Uub))
    <=> ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Uub),Uu)
        & ( size_s7982070591426661849_VEBTi(Uub) = Uua ) ) ) ).

% ATP.lambda_416
tff(fact_10614_ATP_Olambda__417,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_fo(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu))
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_417
tff(fact_10615_ATP_Olambda__418,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( pp(aa_lis9036377640098331692r_bool(aa_nat3680331675976215733r_bool(aTP_Lamm_fm(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu))
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_418
tff(fact_10616_ATP_Olambda__419,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_fn(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_419
tff(fact_10617_ATP_Olambda__420,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_fp(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Uub)),Uu))
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_420
tff(fact_10618_ATP_Olambda__421,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fq(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_421
tff(fact_10619_ATP_Olambda__422,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fr(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_422
tff(fact_10620_ATP_Olambda__423,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_ft(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_423
tff(fact_10621_ATP_Olambda__424,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_alc(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_424
tff(fact_10622_ATP_Olambda__425,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_aav(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Uu),aa_nat_complex(semiri8010041392384452111omplex,Uub))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_425
tff(fact_10623_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_aax(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uub))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_426
tff(fact_10624_ATP_Olambda__427,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_aaw(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uub))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_427
tff(fact_10625_ATP_Olambda__428,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_ee(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uua),Uub)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu),Uub)) ).

% ATP.lambda_428
tff(fact_10626_ATP_Olambda__429,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ald(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_429
tff(fact_10627_ATP_Olambda__430,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_agd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_430
tff(fact_10628_ATP_Olambda__431,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_cc(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua)) ) ) ).

% ATP.lambda_431
tff(fact_10629_ATP_Olambda__432,axiom,
    ! [Uu: product_prod_int_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_set1381271197253378494t_bool(aTP_Lamm_ch(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_432
tff(fact_10630_ATP_Olambda__433,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_cg(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).

% ATP.lambda_433
tff(fact_10631_ATP_Olambda__434,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_cd(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_434
tff(fact_10632_ATP_Olambda__435,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ce(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_435
tff(fact_10633_ATP_Olambda__436,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_cf(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_436
tff(fact_10634_ATP_Olambda__437,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_aht(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_437
tff(fact_10635_ATP_Olambda__438,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_hb(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_438
tff(fact_10636_ATP_Olambda__439,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ahu(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_439
tff(fact_10637_ATP_Olambda__440,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_hd(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_440
tff(fact_10638_ATP_Olambda__441,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_he(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_441
tff(fact_10639_ATP_Olambda__442,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_hc(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_442
tff(fact_10640_ATP_Olambda__443,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ak(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_443
tff(fact_10641_ATP_Olambda__444,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_al(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_444
tff(fact_10642_ATP_Olambda__445,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(aa_set8732061614394825996r_bool(aTP_Lamm_fh(Uu),Uua),Uub))
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_445
tff(fact_10643_ATP_Olambda__446,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: set_complex] :
      ( pp(aa_set_complex_bool(aa_set6592632205191297002x_bool(aTP_Lamm_fi(Uu),Uua),Uub))
    <=> ( minus_811609699411566653omplex(Uub,Uu) = minus_811609699411566653omplex(Uua,Uu) ) ) ).

% ATP.lambda_446
tff(fact_10644_ATP_Olambda__447,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_fj(Uu),Uua),Uub))
    <=> ( minus_minus_set_nat(Uub,Uu) = minus_minus_set_nat(Uua,Uu) ) ) ).

% ATP.lambda_447
tff(fact_10645_ATP_Olambda__448,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( pp(aa_set_int_bool(aa_set7694327796961020398t_bool(aTP_Lamm_fg(Uu),Uua),Uub))
    <=> ( minus_minus_set_int(Uub,Uu) = minus_minus_set_int(Uua,Uu) ) ) ).

% ATP.lambda_448
tff(fact_10646_ATP_Olambda__449,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_au(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_449
tff(fact_10647_ATP_Olambda__450,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_oy(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & pp(aa_Code_integer_bool(Uua,Uub)) ) ) ).

% ATP.lambda_450
tff(fact_10648_ATP_Olambda__451,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_Pr803113274115125473t_bool,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun8917503761599146496t_bool(aTP_Lamm_ay(Uu),Uua),Uub))
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & pp(aa_Pro4501881432599481050t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_451
tff(fact_10649_ATP_Olambda__452,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ax(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_452
tff(fact_10650_ATP_Olambda__453,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_av(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_453
tff(fact_10651_ATP_Olambda__454,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_454
tff(fact_10652_ATP_Olambda__455,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_az(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_455
tff(fact_10653_ATP_Olambda__456,axiom,
    ! [Uu: fun_VEBT_VEBT_bool,Uua: vEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_cn(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_VEBT_VEBT_bool(Uu,Uub)) ) ) ).

% ATP.lambda_456
tff(fact_10654_ATP_Olambda__457,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Pro4501881432599481050t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_457
tff(fact_10655_ATP_Olambda__458,axiom,
    ! [Uu: fun_complex_bool,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_complex_bool(Uu,Uub)) ) ) ).

% ATP.lambda_458
tff(fact_10656_ATP_Olambda__459,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_459
tff(fact_10657_ATP_Olambda__460,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_460
tff(fact_10658_ATP_Olambda__461,axiom,
    ! [Uu: fun_VEBT_VEBT_bool,Uua: vEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_cs(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_VEBT_VEBT_bool(Uu,Uub)) ) ) ).

% ATP.lambda_461
tff(fact_10659_ATP_Olambda__462,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(aTP_Lamm_cu(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Pro4501881432599481050t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_462
tff(fact_10660_ATP_Olambda__463,axiom,
    ! [Uu: fun_complex_bool,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_complex_bool(Uu,Uub)) ) ) ).

% ATP.lambda_463
tff(fact_10661_ATP_Olambda__464,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cv(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_464
tff(fact_10662_ATP_Olambda__465,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cw(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_465
tff(fact_10663_ATP_Olambda__466,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_jf(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_466
tff(fact_10664_ATP_Olambda__467,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_jr(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_467
tff(fact_10665_ATP_Olambda__468,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun5910925830913328781r_bool(aTP_Lamm_jn(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & ( aa_Cod5621350732091595593omplex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_468
tff(fact_10666_ATP_Olambda__469,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_jp(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_469
tff(fact_10667_ATP_Olambda__470,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_jh(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_470
tff(fact_10668_ATP_Olambda__471,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_jt(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_471
tff(fact_10669_ATP_Olambda__472,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_jj(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_472
tff(fact_10670_ATP_Olambda__473,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_jv(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_473
tff(fact_10671_ATP_Olambda__474,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_jl(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_474
tff(fact_10672_ATP_Olambda__475,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_jx(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_475
tff(fact_10673_ATP_Olambda__476,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun437439898845168857T_bool(aTP_Lamm_il(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_476
tff(fact_10674_ATP_Olambda__477,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_ix(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_477
tff(fact_10675_ATP_Olambda__478,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8795547071341169043r_bool(aTP_Lamm_it(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & ( aa_Code_integer_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_478
tff(fact_10676_ATP_Olambda__479,axiom,
    ! [Uu: set_complex,Uua: fun_complex_assn,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1925668587588942193x_bool(aTP_Lamm_iv(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_479
tff(fact_10677_ATP_Olambda__480,axiom,
    ! [Uu: set_real,Uua: fun_real_assn,Uub: real] :
      ( pp(aa_real_bool(aa_fun159070158279343989l_bool(aTP_Lamm_in(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_480
tff(fact_10678_ATP_Olambda__481,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_iz(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_481
tff(fact_10679_ATP_Olambda__482,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1272683333798124589t_bool(aTP_Lamm_ip(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_482
tff(fact_10680_ATP_Olambda__483,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_jb(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_483
tff(fact_10681_ATP_Olambda__484,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: int] :
      ( pp(aa_int_bool(aa_fun1414614411059211637t_bool(aTP_Lamm_ir(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_484
tff(fact_10682_ATP_Olambda__485,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_jd(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_485
tff(fact_10683_ATP_Olambda__486,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: bool] : aa_bool_assn(aa_VEB1972364096475303967l_assn(aTP_Lamm_bl(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uub),aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Uu)))) ).

% ATP.lambda_486
tff(fact_10684_ATP_Olambda__487,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_bv(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),vEBT_vebt_mint(Uu)))) ).

% ATP.lambda_487
tff(fact_10685_ATP_Olambda__488,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_bw(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),vEBT_vebt_maxt(Uu)))) ).

% ATP.lambda_488
tff(fact_10686_ATP_Olambda__489,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: nat] : aa_nat_assn(aa_lis7348694692495225848t_assn(aTP_Lamm_ry(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uu),Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),size_s7982070591426661849_VEBTi(Uua)))) ).

% ATP.lambda_489
tff(fact_10687_ATP_Olambda__490,axiom,
    ! [Uu: array_bool,Uua: list_bool,Uub: nat] : aa_nat_assn(aa_lis1286645363786127574t_assn(aTP_Lamm_rw(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(snga_assn_bool(Uu,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),size_size_list_bool(Uua)))) ).

% ATP.lambda_490
tff(fact_10688_ATP_Olambda__491,axiom,
    ! [Uu: array_nat,Uua: list_nat,Uub: nat] : aa_nat_assn(aa_lis6041139530851742867t_assn(aTP_Lamm_rx(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(snga_assn_nat(Uu,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),size_size_list_nat(Uua)))) ).

% ATP.lambda_491
tff(fact_10689_ATP_Olambda__492,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_lis3169797078229555565i_assn(aTP_Lamm_ik(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uu),Uua)),aa_bool_assn(pure_assn,aa_lis8521252006704567508i_bool(fequal1389750275112723248_VEBTi(Uub),Uua))) ).

% ATP.lambda_492
tff(fact_10690_ATP_Olambda__493,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_ba(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua)) ) ) ).

% ATP.lambda_493
tff(fact_10691_ATP_Olambda__494,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_set1381271197253378494t_bool(aTP_Lamm_be(Uu),Uua),Uub))
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_494
tff(fact_10692_ATP_Olambda__495,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_bd(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ~ pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).

% ATP.lambda_495
tff(fact_10693_ATP_Olambda__496,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_bb(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_496
tff(fact_10694_ATP_Olambda__497,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_497
tff(fact_10695_ATP_Olambda__498,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_bc(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_498
tff(fact_10696_ATP_Olambda__499,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aig(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_499
tff(fact_10697_ATP_Olambda__500,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_hp(Uu),Uua),Uub))
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_500
tff(fact_10698_ATP_Olambda__501,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_ahn(Uu),Uua),Uub))
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_501
tff(fact_10699_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_akh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_502
tff(fact_10700_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_503
tff(fact_10701_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_adh(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(Uu,Uub)),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uub)) ).

% ATP.lambda_504
tff(fact_10702_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_acx(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_505
tff(fact_10703_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_506
tff(fact_10704_ATP_Olambda__507,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_adi(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_507
tff(fact_10705_ATP_Olambda__508,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_adu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_508
tff(fact_10706_ATP_Olambda__509,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_adj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_509
tff(fact_10707_ATP_Olambda__510,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fk(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_510
tff(fact_10708_ATP_Olambda__511,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_511
tff(fact_10709_ATP_Olambda__512,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aks(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_512
tff(fact_10710_ATP_Olambda__513,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ro(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_513
tff(fact_10711_ATP_Olambda__514,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_rm(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_514
tff(fact_10712_ATP_Olambda__515,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_515
tff(fact_10713_ATP_Olambda__516,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_rn(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_516
tff(fact_10714_ATP_Olambda__517,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ym(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_517
tff(fact_10715_ATP_Olambda__518,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ne(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_518
tff(fact_10716_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_519
tff(fact_10717_ATP_Olambda__520,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ng(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_520
tff(fact_10718_ATP_Olambda__521,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_sg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_VEBT_VEBT_nat(Uua,Uub)),aa_VEBT_VEBT_nat(Uu,Uub)) ).

% ATP.lambda_521
tff(fact_10719_ATP_Olambda__522,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_sa(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_522
tff(fact_10720_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ut(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_523
tff(fact_10721_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_524
tff(fact_10722_ATP_Olambda__525,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_rz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_525
tff(fact_10723_ATP_Olambda__526,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_op(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_526
tff(fact_10724_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_527
tff(fact_10725_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_oo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_528
tff(fact_10726_ATP_Olambda__529,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_vg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_529
tff(fact_10727_ATP_Olambda__530,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_or(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_530
tff(fact_10728_ATP_Olambda__531,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ajf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_531
tff(fact_10729_ATP_Olambda__532,axiom,
    ! [Uu: fun_Co8822979462182099027r_bool,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_gu(Uu),Uua),Uub))
    <=> ( pp(aa_Code_integer_bool(Uu,Uub))
        | pp(aa_Code_integer_bool(Uua,Uub)) ) ) ).

% ATP.lambda_532
tff(fact_10730_ATP_Olambda__533,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: fun_Pr803113274115125473t_bool,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun8917503761599146496t_bool(aTP_Lamm_gr(Uu),Uua),Uub))
    <=> ( pp(aa_Pro4501881432599481050t_bool(Uu,Uub))
        | pp(aa_Pro4501881432599481050t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_533
tff(fact_10731_ATP_Olambda__534,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_gv(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_534
tff(fact_10732_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_gs(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_535
tff(fact_10733_ATP_Olambda__536,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_gt(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_536
tff(fact_10734_ATP_Olambda__537,axiom,
    ! [Uu: fun_Co8822979462182099027r_bool,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_gz(Uu),Uua),Uub))
    <=> ( pp(aa_Code_integer_bool(Uu,Uub))
        & pp(aa_Code_integer_bool(Uua,Uub)) ) ) ).

% ATP.lambda_537
tff(fact_10735_ATP_Olambda__538,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: fun_Pr803113274115125473t_bool,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun8917503761599146496t_bool(aTP_Lamm_gw(Uu),Uua),Uub))
    <=> ( pp(aa_Pro4501881432599481050t_bool(Uu,Uub))
        & pp(aa_Pro4501881432599481050t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_538
tff(fact_10736_ATP_Olambda__539,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ha(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_539
tff(fact_10737_ATP_Olambda__540,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_gx(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_540
tff(fact_10738_ATP_Olambda__541,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_gy(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_541
tff(fact_10739_ATP_Olambda__542,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_yk(Uu),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_542
tff(fact_10740_ATP_Olambda__543,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_ya(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_543
tff(fact_10741_ATP_Olambda__544,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_yd(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_544
tff(fact_10742_ATP_Olambda__545,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_yg(Uu),Uua),Uub) = groups705719431365010083at_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_545
tff(fact_10743_ATP_Olambda__546,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_yi(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_546
tff(fact_10744_ATP_Olambda__547,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_oa(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_547
tff(fact_10745_ATP_Olambda__548,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_od(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_548
tff(fact_10746_ATP_Olambda__549,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_nx(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_549
tff(fact_10747_ATP_Olambda__550,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_og(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_550
tff(fact_10748_ATP_Olambda__551,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fl(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_551
tff(fact_10749_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_os(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_552
tff(fact_10750_ATP_Olambda__553,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_zy(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_553
tff(fact_10751_ATP_Olambda__554,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ot(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_554
tff(fact_10752_ATP_Olambda__555,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_oj(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_555
tff(fact_10753_ATP_Olambda__556,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vi(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_556
tff(fact_10754_ATP_Olambda__557,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ok(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_557
tff(fact_10755_ATP_Olambda__558,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_bp(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(Uu,Uub)),Uua) ).

% ATP.lambda_558
tff(fact_10756_ATP_Olambda__559,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_nm(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_559
tff(fact_10757_ATP_Olambda__560,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_br(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_option_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_560
tff(fact_10758_ATP_Olambda__561,axiom,
    ! [Uu: fun_li3352006551604695530i_assn,Uua: assn,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_ass1897699608592665680i_assn(aTP_Lamm_eh(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(Uu,Uub)),Uua) ).

% ATP.lambda_561
tff(fact_10759_ATP_Olambda__562,axiom,
    ! [Uu: fun_bool_assn,Uua: assn,Uub: bool] : aa_bool_assn(aa_ass2376637134808068204l_assn(aTP_Lamm_bq(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_bool_assn(Uu,Uub)),Uua) ).

% ATP.lambda_562
tff(fact_10760_ATP_Olambda__563,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_bs(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_563
tff(fact_10761_ATP_Olambda__564,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_564
tff(fact_10762_ATP_Olambda__565,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_565
tff(fact_10763_ATP_Olambda__566,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_no(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_566
tff(fact_10764_ATP_Olambda__567,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vw(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_567
tff(fact_10765_ATP_Olambda__568,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_568
tff(fact_10766_ATP_Olambda__569,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_569
tff(fact_10767_ATP_Olambda__570,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_sn(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uub)),Uua) ).

% ATP.lambda_570
tff(fact_10768_ATP_Olambda__571,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_so(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_571
tff(fact_10769_ATP_Olambda__572,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ajc(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_572
tff(fact_10770_ATP_Olambda__573,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_573
tff(fact_10771_ATP_Olambda__574,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_yn(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_574
tff(fact_10772_ATP_Olambda__575,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_yo(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_575
tff(fact_10773_ATP_Olambda__576,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_akm(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_576
tff(fact_10774_ATP_Olambda__577,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aje(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_577
tff(fact_10775_ATP_Olambda__578,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_ci(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_578
tff(fact_10776_ATP_Olambda__579,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr803113274115125473t_bool,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun8917503761599146496t_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_Pro4501881432599481050t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_579
tff(fact_10777_ATP_Olambda__580,axiom,
    ! [Uu: complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_580
tff(fact_10778_ATP_Olambda__581,axiom,
    ! [Uu: nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_581
tff(fact_10779_ATP_Olambda__582,axiom,
    ! [Uu: int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ck(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_582
tff(fact_10780_ATP_Olambda__583,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_agl(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub))) ).

% ATP.lambda_583
tff(fact_10781_ATP_Olambda__584,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_agn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub))) ).

% ATP.lambda_584
tff(fact_10782_ATP_Olambda__585,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_agm(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub))) ).

% ATP.lambda_585
tff(fact_10783_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abq(Uu),Uua),Uub) = groups708209901874060359at_nat(Uu,set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_586
tff(fact_10784_ATP_Olambda__587,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_abp(Uu),Uua),Uub) = groups705719431365010083at_int(Uu,set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_587
tff(fact_10785_ATP_Olambda__588,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sr(Uu),Uua),Uub) = groups6591440286371151544t_real(Uu,set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_588
tff(fact_10786_ATP_Olambda__589,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sq(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_589
tff(fact_10787_ATP_Olambda__590,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rg(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_590
tff(fact_10788_ATP_Olambda__591,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_591
tff(fact_10789_ATP_Olambda__592,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_rh(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_592
tff(fact_10790_ATP_Olambda__593,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_593
tff(fact_10791_ATP_Olambda__594,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ri(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_594
tff(fact_10792_ATP_Olambda__595,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_tn(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_595
tff(fact_10793_ATP_Olambda__596,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_tm(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_596
tff(fact_10794_ATP_Olambda__597,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tq(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_597
tff(fact_10795_ATP_Olambda__598,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_to(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_598
tff(fact_10796_ATP_Olambda__599,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tp(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_599
tff(fact_10797_ATP_Olambda__600,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_akb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_600
tff(fact_10798_ATP_Olambda__601,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_pu(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_601
tff(fact_10799_ATP_Olambda__602,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pt(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_602
tff(fact_10800_ATP_Olambda__603,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_px(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_603
tff(fact_10801_ATP_Olambda__604,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_pv(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_604
tff(fact_10802_ATP_Olambda__605,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pw(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_605
tff(fact_10803_ATP_Olambda__606,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_606
tff(fact_10804_ATP_Olambda__607,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_akx(Uu),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua))) ) ).

% ATP.lambda_607
tff(fact_10805_ATP_Olambda__608,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ni(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_608
tff(fact_10806_ATP_Olambda__609,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_609
tff(fact_10807_ATP_Olambda__610,axiom,
    ! [Uu: assn,Uua: fun_li3352006551604695530i_assn,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_fun6795544665460849810i_assn(aTP_Lamm_eg(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_lis2508312872377092907i_assn(Uua,Uub)) ).

% ATP.lambda_610
tff(fact_10808_ATP_Olambda__611,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_611
tff(fact_10809_ATP_Olambda__612,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_612
tff(fact_10810_ATP_Olambda__613,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_nk(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_613
tff(fact_10811_ATP_Olambda__614,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_ea(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_VEBT_VEBTi_assn(Uu,Uub)) ).

% ATP.lambda_614
tff(fact_10812_ATP_Olambda__615,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_ec(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_option_nat_assn(Uu,Uub)) ).

% ATP.lambda_615
tff(fact_10813_ATP_Olambda__616,axiom,
    ! [Uu: fun_bool_assn,Uua: assn,Uub: bool] : aa_bool_assn(aa_ass2376637134808068204l_assn(aTP_Lamm_eb(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_bool_assn(Uu,Uub)) ).

% ATP.lambda_616
tff(fact_10814_ATP_Olambda__617,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_ed(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_nat_assn(Uu,Uub)) ).

% ATP.lambda_617
tff(fact_10815_ATP_Olambda__618,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_618
tff(fact_10816_ATP_Olambda__619,axiom,
    ! [Uu: code_integer,Uua: fun_nat_nat,Uub: nat] : aa_nat_Code_integer(aa_fun8101275520843608965nteger(aTP_Lamm_aaq(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_619
tff(fact_10817_ATP_Olambda__620,axiom,
    ! [Uu: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_aap(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_620
tff(fact_10818_ATP_Olambda__621,axiom,
    ! [Uu: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_aao(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_621
tff(fact_10819_ATP_Olambda__622,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aat(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_622
tff(fact_10820_ATP_Olambda__623,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_aar(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_623
tff(fact_10821_ATP_Olambda__624,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_aas(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_624
tff(fact_10822_ATP_Olambda__625,axiom,
    ! [Uu: assn,Uua: vEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(aTP_Lamm_dw(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Uub),Uua))) ).

% ATP.lambda_625
tff(fact_10823_ATP_Olambda__626,axiom,
    ! [Uu: assn,Uua: option_nat,Uub: option_nat] : aa_option_nat_assn(aa_opt3861602675778688675t_assn(aTP_Lamm_dy(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),Uua))) ).

% ATP.lambda_626
tff(fact_10824_ATP_Olambda__627,axiom,
    ! [Uu: assn,Uua: bool,Uub: bool] : aa_bool_assn(aa_boo2938360491098410167l_assn(aTP_Lamm_dx(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uub),Uua))) ).

% ATP.lambda_627
tff(fact_10825_ATP_Olambda__628,axiom,
    ! [Uu: assn,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_dz(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uua))) ).

% ATP.lambda_628
tff(fact_10826_ATP_Olambda__629,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_acg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_629
tff(fact_10827_ATP_Olambda__630,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_acf(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_630
tff(fact_10828_ATP_Olambda__631,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aci(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_631
tff(fact_10829_ATP_Olambda__632,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ach(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_632
tff(fact_10830_ATP_Olambda__633,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aTP_Lamm_id(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uua)),aa_nat_nat(suc,aa_Code_integer_nat(Uu,Uub))),aa_Code_integer_nat(Uu,Uub)))) ) ).

% ATP.lambda_633
tff(fact_10831_ATP_Olambda__634,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(aTP_Lamm_ia(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Pro4501881432599481050t_bool(fequal6010348034681699726nt_int(Uub),Uua)),aa_nat_nat(suc,aa_Pro6801748904974264397nt_nat(Uu,Uub))),aa_Pro6801748904974264397nt_nat(Uu,Uub)))) ) ).

% ATP.lambda_634
tff(fact_10832_ATP_Olambda__635,axiom,
    ! [Uu: fun_complex_nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ie(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uua)),aa_nat_nat(suc,aa_complex_nat(Uu,Uub))),aa_complex_nat(Uu,Uub)))) ) ).

% ATP.lambda_635
tff(fact_10833_ATP_Olambda__636,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ib(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uua)),aa_nat_nat(suc,aa_nat_nat(Uu,Uub))),aa_nat_nat(Uu,Uub)))) ) ).

% ATP.lambda_636
tff(fact_10834_ATP_Olambda__637,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ic(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uua)),aa_nat_nat(suc,aa_int_nat(Uu,Uub))),aa_int_nat(Uu,Uub)))) ) ).

% ATP.lambda_637
tff(fact_10835_ATP_Olambda__638,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_ii(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(Uua,Uub)),aa_Code_integer_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_638
tff(fact_10836_ATP_Olambda__639,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr803113274115125473t_bool,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun8917503761599146496t_bool(aTP_Lamm_if(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Pro4501881432599481050t_bool(Uua,Uub)),aa_Pro6801748904974264397nt_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_639
tff(fact_10837_ATP_Olambda__640,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ij(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(Uua,Uub)),aa_complex_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_640
tff(fact_10838_ATP_Olambda__641,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ig(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uua,Uub)),aa_nat_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_641
tff(fact_10839_ATP_Olambda__642,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ih(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uua,Uub)),aa_int_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_642
tff(fact_10840_ATP_Olambda__643,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun3540204951715082475r_bool(aTP_Lamm_hx(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub)))) ) ).

% ATP.lambda_643
tff(fact_10841_ATP_Olambda__644,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5604724294258851521t_bool(aTP_Lamm_hr(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uua,Uub)))) ) ).

% ATP.lambda_644
tff(fact_10842_ATP_Olambda__645,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aTP_Lamm_hz(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub)))) ) ).

% ATP.lambda_645
tff(fact_10843_ATP_Olambda__646,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aTP_Lamm_ht(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)))) ) ).

% ATP.lambda_646
tff(fact_10844_ATP_Olambda__647,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aTP_Lamm_hv(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)))) ) ).

% ATP.lambda_647
tff(fact_10845_ATP_Olambda__648,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_si(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_648
tff(fact_10846_ATP_Olambda__649,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sh(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_649
tff(fact_10847_ATP_Olambda__650,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_abi(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_650
tff(fact_10848_ATP_Olambda__651,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_akv(Uu),Uua),Uub))
    <=> pp(aa_real_bool(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua))) ) ).

% ATP.lambda_651
tff(fact_10849_ATP_Olambda__652,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ps(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_652
tff(fact_10850_ATP_Olambda__653,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aku(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_653
tff(fact_10851_ATP_Olambda__654,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_654
tff(fact_10852_ATP_Olambda__655,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_aau(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_655
tff(fact_10853_ATP_Olambda__656,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vz(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_656
tff(fact_10854_ATP_Olambda__657,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_wa(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_657
tff(fact_10855_ATP_Olambda__658,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun9093735442950272378T_real(aTP_Lamm_abw(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_VEBT_VEBT_complex(Uu,Uub)),aa_VEBT_VEBT_complex(Uua,Uub))) ).

% ATP.lambda_658
tff(fact_10856_ATP_Olambda__659,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_abt(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_VEBT_VEBT_real(Uu,Uub)),aa_VEBT_VEBT_real(Uua,Uub))) ).

% ATP.lambda_659
tff(fact_10857_ATP_Olambda__660,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_abx(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_real_complex(Uu,Uub)),aa_real_complex(Uua,Uub))) ).

% ATP.lambda_660
tff(fact_10858_ATP_Olambda__661,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abu(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_661
tff(fact_10859_ATP_Olambda__662,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_aca(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub))) ).

% ATP.lambda_662
tff(fact_10860_ATP_Olambda__663,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_abz(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_663
tff(fact_10861_ATP_Olambda__664,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_aby(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_int_complex(Uu,Uub)),aa_int_complex(Uua,Uub))) ).

% ATP.lambda_664
tff(fact_10862_ATP_Olambda__665,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_abv(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_665
tff(fact_10863_ATP_Olambda__666,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_akg(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_666
tff(fact_10864_ATP_Olambda__667,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_di(Uu),Uua),Uub) = aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua))) ).

% ATP.lambda_667
tff(fact_10865_ATP_Olambda__668,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_dh(Uu),Uua),Uub) = aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua))) ).

% ATP.lambda_668
tff(fact_10866_ATP_Olambda__669,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_adx(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_complex(Uua,Uuc)),aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_complex),aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_669
tff(fact_10867_ATP_Olambda__670,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aef(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_670
tff(fact_10868_ATP_Olambda__671,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_adz(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_671
tff(fact_10869_ATP_Olambda__672,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aed(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_672
tff(fact_10870_ATP_Olambda__673,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aeb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_int(Uua,Uuc)),aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_673
tff(fact_10871_ATP_Olambda__674,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] : aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aeh(Uu),Uua),Uub),Uuc) = if_assn(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_assn(Uua,Uuc),if_assn(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_assn,aa_nat_assn(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_674
tff(fact_10872_ATP_Olambda__675,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aej(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_675
tff(fact_10873_ATP_Olambda__676,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aek(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_676
tff(fact_10874_ATP_Olambda__677,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aem(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_677
tff(fact_10875_ATP_Olambda__678,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ael(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_int(Uua,Uuc)),aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_678
tff(fact_10876_ATP_Olambda__679,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ady(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_679
tff(fact_10877_ATP_Olambda__680,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] : aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_aei(Uu),Uua),Uub),Uuc) = if_assn(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ).

% ATP.lambda_680
tff(fact_10878_ATP_Olambda__681,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_aeg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_681
tff(fact_10879_ATP_Olambda__682,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aea(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_682
tff(fact_10880_ATP_Olambda__683,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_aee(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_683
tff(fact_10881_ATP_Olambda__684,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aec(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_684
tff(fact_10882_ATP_Olambda__685,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_uu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uuc),Uua)),aa_nat_real(Uub,Uuc)),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_685
tff(fact_10883_ATP_Olambda__686,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_qa(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu)),aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_686
tff(fact_10884_ATP_Olambda__687,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_qe(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu),aa_VEBT_VEBT_rat(Uua,Uuc)),aa_VEBT_VEBT_rat(Uub,Uuc)) ).

% ATP.lambda_687
tff(fact_10885_ATP_Olambda__688,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_qi(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu)),aa_VEBT_VEBT_nat(Uua,Uuc)),aa_VEBT_VEBT_nat(Uub,Uuc)) ).

% ATP.lambda_688
tff(fact_10886_ATP_Olambda__689,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_qc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uuc),Uu)),aa_Code_integer_real(Uua,Uuc)),aa_Code_integer_real(Uub,Uuc)) ).

% ATP.lambda_689
tff(fact_10887_ATP_Olambda__690,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] : aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_qg(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uuc),Uu),aa_Code_integer_rat(Uua,Uuc)),aa_Code_integer_rat(Uub,Uuc)) ).

% ATP.lambda_690
tff(fact_10888_ATP_Olambda__691,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_qd(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_691
tff(fact_10889_ATP_Olambda__692,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_qh(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_692
tff(fact_10890_ATP_Olambda__693,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_qb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_693
tff(fact_10891_ATP_Olambda__694,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_qf(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_694
tff(fact_10892_ATP_Olambda__695,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] : aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_qj(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_695
tff(fact_10893_ATP_Olambda__696,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_abs(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_696
tff(fact_10894_ATP_Olambda__697,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_abr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_697
tff(fact_10895_ATP_Olambda__698,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ru(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_698
tff(fact_10896_ATP_Olambda__699,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rt(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rs(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_699
tff(fact_10897_ATP_Olambda__700,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rr(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rq(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_700
tff(fact_10898_ATP_Olambda__701,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_nw(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_nv(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_701
tff(fact_10899_ATP_Olambda__702,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_nu(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_nt(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_702
tff(fact_10900_ATP_Olambda__703,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ns(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_nr(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_703
tff(fact_10901_ATP_Olambda__704,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_nq(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_np(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_704
tff(fact_10902_ATP_Olambda__705,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_zt(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_zr(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_zs(Uu),Uub),Uuc))) ).

% ATP.lambda_705
tff(fact_10903_ATP_Olambda__706,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8197081679332685312t_bool,Uuc: int] : aa_int_int(aa_fun7408359038690759577nt_int(aa_fun3657818423959754427nt_int(aTP_Lamm_zo(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_int1390634949194299888er_int(aTP_Lamm_zm(Uua),Uuc),collect_Code_integer(aa_int3865899974016033897r_bool(aa_fun4269100936724824892r_bool(aTP_Lamm_zn(Uu),Uub),Uuc))) ).

% ATP.lambda_706
tff(fact_10904_ATP_Olambda__707,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_zj(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_zi(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_hi(Uu),Uub),Uuc))) ).

% ATP.lambda_707
tff(fact_10905_ATP_Olambda__708,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: int] : aa_int_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_zf(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_int1343382402272391561BT_int(aTP_Lamm_ze(Uua),Uuc),collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_hh(Uu),Uub),Uuc))) ).

% ATP.lambda_708
tff(fact_10906_ATP_Olambda__709,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_zb(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_za(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_mf(Uu),Uub),Uuc))) ).

% ATP.lambda_709
tff(fact_10907_ATP_Olambda__710,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co822469698833053651at_int,Uub: fun_Co8657364630979331932t_bool,Uuc: nat] : aa_nat_int(aa_fun3371163242325571225at_int(aa_fun5074241136823302423at_int(aTP_Lamm_yy(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_nat1674686023564409748er_int(aTP_Lamm_yx(Uua),Uuc),collect_Code_integer(aa_nat6120813132205607877r_bool(aa_fun8839232882156773436r_bool(aTP_Lamm_ma(Uu),Uub),Uuc))) ).

% ATP.lambda_710
tff(fact_10908_ATP_Olambda__711,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_yv(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_yu(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_hg(Uu),Uub),Uuc))) ).

% ATP.lambda_711
tff(fact_10909_ATP_Olambda__712,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_int(aa_fun1384843155787029862at_int(aa_fun6066863168309489065at_int(aTP_Lamm_ys(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_yr(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_hf(Uu),Uub),Uuc))) ).

% ATP.lambda_712
tff(fact_10910_ATP_Olambda__713,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_zx(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_lq(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_hg(Uu),Uub),Uuc))) ).

% ATP.lambda_713
tff(fact_10911_ATP_Olambda__714,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_zv(Uu),Uua),Uub),Uuc) = groups6361806394783013919BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_lm(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_hf(Uu),Uub),Uuc))) ).

% ATP.lambda_714
tff(fact_10912_ATP_Olambda__715,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co7124838579184476414x_bool,Uuc: complex] : aa_complex_complex(aa_fun8169706760919072411omplex(aa_fun2334956106851614393omplex(aTP_Lamm_nd(Uu),Uua),Uub),Uuc) = groups8024822376189712711omplex(aa_com7552888802784664052omplex(aTP_Lamm_nb(Uua),Uuc),collect_Code_integer(aa_com339898768654300263r_bool(aa_fun8191597921216526780r_bool(aTP_Lamm_nc(Uu),Uub),Uuc))) ).

% ATP.lambda_715
tff(fact_10913_ATP_Olambda__716,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: complex] : aa_complex_complex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_my(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_mw(Uua),Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_mx(Uu),Uub),Uuc))) ).

% ATP.lambda_716
tff(fact_10914_ATP_Olambda__717,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mr(Uua),Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_ms(Uu),Uub),Uuc))) ).

% ATP.lambda_717
tff(fact_10915_ATP_Olambda__718,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_mo(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_mn(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_hm(Uu),Uub),Uuc))) ).

% ATP.lambda_718
tff(fact_10916_ATP_Olambda__719,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: complex] : aa_complex_complex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_mk(Uu),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_mj(Uua),Uuc),collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_hl(Uu),Uub),Uuc))) ).

% ATP.lambda_719
tff(fact_10917_ATP_Olambda__720,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_me(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_mf(Uu),Uub),Uuc))) ).

% ATP.lambda_720
tff(fact_10918_ATP_Olambda__721,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co8657364630979331932t_bool,Uuc: nat] : aa_nat_nat(aa_fun7549014261834767933at_nat(aa_fun5721305349739569687at_nat(aTP_Lamm_mb(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_lz(Uua),Uuc),collect_Code_integer(aa_nat6120813132205607877r_bool(aa_fun8839232882156773436r_bool(aTP_Lamm_ma(Uu),Uub),Uuc))) ).

% ATP.lambda_721
tff(fact_10919_ATP_Olambda__722,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_lw(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_lu(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_lv(Uu),Uub),Uuc))) ).

% ATP.lambda_722
tff(fact_10920_ATP_Olambda__723,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_lr(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_lq(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_hg(Uu),Uub),Uuc))) ).

% ATP.lambda_723
tff(fact_10921_ATP_Olambda__724,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_ln(Uu),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_lm(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_hf(Uu),Uub),Uuc))) ).

% ATP.lambda_724
tff(fact_10922_ATP_Olambda__725,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_adw(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_adv(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_725
tff(fact_10923_ATP_Olambda__726,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_adt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ads(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_726
tff(fact_10924_ATP_Olambda__727,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_adr(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_adq(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),Uuc)) ).

% ATP.lambda_727
tff(fact_10925_ATP_Olambda__728,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_adp(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ado(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_728
tff(fact_10926_ATP_Olambda__729,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_fun8469111845059130348nteger(aTP_Lamm_adn(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_fun1713859001248219487nteger(aTP_Lamm_adm(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),Uuc)) ).

% ATP.lambda_729
tff(fact_10927_ATP_Olambda__730,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_adl(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_adk(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_730
tff(fact_10928_ATP_Olambda__731,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ajm(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_731
tff(fact_10929_ATP_Olambda__732,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ajk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_732
tff(fact_10930_ATP_Olambda__733,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ajj(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_733
tff(fact_10931_ATP_Olambda__734,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_aji(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_734
tff(fact_10932_ATP_Olambda__735,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_su(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uub)) ).

% ATP.lambda_735
tff(fact_10933_ATP_Olambda__736,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ss(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uub)) ).

% ATP.lambda_736
tff(fact_10934_ATP_Olambda__737,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ta(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_737
tff(fact_10935_ATP_Olambda__738,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_sw(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_738
tff(fact_10936_ATP_Olambda__739,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_sy(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uub)) ).

% ATP.lambda_739
tff(fact_10937_ATP_Olambda__740,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rq(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uua),Uu)),Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_740
tff(fact_10938_ATP_Olambda__741,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ru(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_741
tff(fact_10939_ATP_Olambda__742,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rs(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_742
tff(fact_10940_ATP_Olambda__743,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_agq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_743
tff(fact_10941_ATP_Olambda__744,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ago(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_744
tff(fact_10942_ATP_Olambda__745,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_agp(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_745
tff(fact_10943_ATP_Olambda__746,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_agh(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_746
tff(fact_10944_ATP_Olambda__747,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_agg(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_747
tff(fact_10945_ATP_Olambda__748,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_agk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_748
tff(fact_10946_ATP_Olambda__749,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_agi(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_749
tff(fact_10947_ATP_Olambda__750,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_agf(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_750
tff(fact_10948_ATP_Olambda__751,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_agj(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_751
tff(fact_10949_ATP_Olambda__752,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_td(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)) ).

% ATP.lambda_752
tff(fact_10950_ATP_Olambda__753,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_tc(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)) ).

% ATP.lambda_753
tff(fact_10951_ATP_Olambda__754,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_754
tff(fact_10952_ATP_Olambda__755,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_te(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_755
tff(fact_10953_ATP_Olambda__756,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tf(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)) ).

% ATP.lambda_756
tff(fact_10954_ATP_Olambda__757,axiom,
    ! [Uu: set_complex,Uua: fun_VE2128610149277289041x_bool,Uub: vEBT_VEBT,Uuc: complex] :
      ( pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_mh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_757
tff(fact_10955_ATP_Olambda__758,axiom,
    ! [Uu: set_complex,Uua: fun_Co7124838579184476414x_bool,Uub: code_integer,Uuc: complex] :
      ( pp(aa_complex_bool(aa_Cod6819470024986797159x_bool(aa_fun3619973020723243580x_bool(aTP_Lamm_mz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_Cod6819470024986797159x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_758
tff(fact_10956_ATP_Olambda__759,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ml(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_759
tff(fact_10957_ATP_Olambda__760,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_mp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_760
tff(fact_10958_ATP_Olambda__761,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_mu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_761
tff(fact_10959_ATP_Olambda__762,axiom,
    ! [Uu: set_nat,Uua: fun_VE3763956218935105199t_bool,Uub: vEBT_VEBT,Uuc: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_lk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_762
tff(fact_10960_ATP_Olambda__763,axiom,
    ! [Uu: set_nat,Uua: fun_Co8657364630979331932t_bool,Uub: code_integer,Uuc: nat] :
      ( pp(aa_nat_bool(aa_Cod2498268374540730821t_bool(aa_fun498830667440607164t_bool(aTP_Lamm_lx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_Cod2498268374540730821t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_763
tff(fact_10961_ATP_Olambda__764,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_mc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_764
tff(fact_10962_ATP_Olambda__765,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_lo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_765
tff(fact_10963_ATP_Olambda__766,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ls(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_766
tff(fact_10964_ATP_Olambda__767,axiom,
    ! [Uu: set_int,Uua: fun_VE3303673267288458579t_bool,Uub: vEBT_VEBT,Uuc: int] :
      ( pp(aa_int_bool(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_zc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_767
tff(fact_10965_ATP_Olambda__768,axiom,
    ! [Uu: set_int,Uua: fun_Co8197081679332685312t_bool,Uub: code_integer,Uuc: int] :
      ( pp(aa_int_bool(aa_Cod8597349444926882409t_bool(aa_fun6946700965406361788t_bool(aTP_Lamm_zk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_Cod8597349444926882409t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_768
tff(fact_10966_ATP_Olambda__769,axiom,
    ! [Uu: set_int,Uua: fun_co2381122952212249071t_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_zp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_769
tff(fact_10967_ATP_Olambda__770,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_zg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_770
tff(fact_10968_ATP_Olambda__771,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE1123255499321568226r_bool,Uub: code_integer,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_Cod3526525079615536833T_bool(aa_fun7592865282689444384T_bool(aTP_Lamm_hj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_Code_integer_bool(aa_VEB6607917098650726363r_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_771
tff(fact_10969_ATP_Olambda__772,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2128610149277289041x_bool,Uub: complex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_hl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_772
tff(fact_10970_ATP_Olambda__773,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3763956218935105199t_bool,Uub: nat,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_hf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_773
tff(fact_10971_ATP_Olambda__774,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3303673267288458579t_bool,Uub: int,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_hh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_774
tff(fact_10972_ATP_Olambda__775,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co7124838579184476414x_bool,Uub: complex,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_com339898768654300263r_bool(aa_fun8191597921216526780r_bool(aTP_Lamm_nc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & pp(aa_complex_bool(aa_Cod6819470024986797159x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_775
tff(fact_10973_ATP_Olambda__776,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8657364630979331932t_bool,Uub: nat,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_nat6120813132205607877r_bool(aa_fun8839232882156773436r_bool(aTP_Lamm_ma(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & pp(aa_nat_bool(aa_Cod2498268374540730821t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_776
tff(fact_10974_ATP_Olambda__777,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8197081679332685312t_bool,Uub: int,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_int3865899974016033897r_bool(aa_fun4269100936724824892r_bool(aTP_Lamm_zn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & pp(aa_int_bool(aa_Cod8597349444926882409t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_777
tff(fact_10975_ATP_Olambda__778,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_mf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_778
tff(fact_10976_ATP_Olambda__779,axiom,
    ! [Uu: set_complex,Uua: fun_co2381122952212249071t_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_zs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_779
tff(fact_10977_ATP_Olambda__780,axiom,
    ! [Uu: set_real,Uua: fun_re1120004649140997760r_bool,Uub: code_integer,Uuc: real] :
      ( pp(aa_real_bool(aa_Cod181991726960451049l_bool(aa_fun8452463775060891964l_bool(aTP_Lamm_hk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_Code_integer_bool(aa_rea7928672538722535017r_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_780
tff(fact_10978_ATP_Olambda__781,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_hm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_781
tff(fact_10979_ATP_Olambda__782,axiom,
    ! [Uu: set_real,Uua: fun_re9121486324101951821t_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_hg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_782
tff(fact_10980_ATP_Olambda__783,axiom,
    ! [Uu: set_real,Uua: fun_re8661203372455305201t_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_hi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_783
tff(fact_10981_ATP_Olambda__784,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_ms(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_784
tff(fact_10982_ATP_Olambda__785,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_hn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_785
tff(fact_10983_ATP_Olambda__786,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_ho(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_786
tff(fact_10984_ATP_Olambda__787,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_mx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_787
tff(fact_10985_ATP_Olambda__788,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_lv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_788
tff(fact_10986_ATP_Olambda__789,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_sv(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_789
tff(fact_10987_ATP_Olambda__790,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_st(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_790
tff(fact_10988_ATP_Olambda__791,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_791
tff(fact_10989_ATP_Olambda__792,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_sx(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_792
tff(fact_10990_ATP_Olambda__793,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_sz(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_793
tff(fact_10991_ATP_Olambda__794,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_ti(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_794
tff(fact_10992_ATP_Olambda__795,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_th(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_795
tff(fact_10993_ATP_Olambda__796,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tl(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_796
tff(fact_10994_ATP_Olambda__797,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_tj(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_797
tff(fact_10995_ATP_Olambda__798,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tk(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_798
tff(fact_10996_ATP_Olambda__799,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_age(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_799
tff(fact_10997_ATP_Olambda__800,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_ail(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_800
tff(fact_10998_ATP_Olambda__801,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_aij(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_801
tff(fact_10999_ATP_Olambda__802,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: fun_VEBT_VEBT_assn,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun437439898845168857T_bool(aa_fun2621268342929013301T_bool(aTP_Lamm_im(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(Uua,Uuc)),aa_VEBT_VEBT_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_802
tff(fact_11000_ATP_Olambda__803,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_iy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_803
tff(fact_11001_ATP_Olambda__804,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8795547071341169043r_bool(aa_fun7758107644544034538r_bool(aTP_Lamm_iu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & ( aa_assn_assn(times_times_assn(aa_Code_integer_assn(Uua,Uuc)),aa_Code_integer_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_804
tff(fact_11002_ATP_Olambda__805,axiom,
    ! [Uu: set_complex,Uua: fun_complex_assn,Uub: fun_complex_assn,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun1925668587588942193x_bool(aa_fun8862831692315842871x_bool(aTP_Lamm_iw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_assn_assn(times_times_assn(aa_complex_assn(Uua,Uuc)),aa_complex_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_805
tff(fact_11003_ATP_Olambda__806,axiom,
    ! [Uu: set_real,Uua: fun_real_assn,Uub: fun_real_assn,Uuc: real] :
      ( pp(aa_real_bool(aa_fun159070158279343989l_bool(aa_fun6385136150007740861l_bool(aTP_Lamm_io(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_assn_assn(times_times_assn(aa_real_assn(Uua,Uuc)),aa_real_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_806
tff(fact_11004_ATP_Olambda__807,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ja(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_807
tff(fact_11005_ATP_Olambda__808,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1272683333798124589t_bool(aa_fun7103831392407232081t_bool(aTP_Lamm_iq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_assn_assn(times_times_assn(aa_nat_assn(Uua,Uuc)),aa_nat_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_808
tff(fact_11006_ATP_Olambda__809,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_jc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_809
tff(fact_11007_ATP_Olambda__810,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1414614411059211637t_bool(aa_fun5523883950577104445t_bool(aTP_Lamm_is(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_assn_assn(times_times_assn(aa_int_assn(Uua,Uuc)),aa_int_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_810
tff(fact_11008_ATP_Olambda__811,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_je(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_811
tff(fact_11009_ATP_Olambda__812,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_jg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_812
tff(fact_11010_ATP_Olambda__813,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_js(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_813
tff(fact_11011_ATP_Olambda__814,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: fun_Co1900716531650368264omplex,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun5910925830913328781r_bool(aa_fun6901856055265004766r_bool(aTP_Lamm_jo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_Cod5621350732091595593omplex(Uua,Uuc)),aa_Cod5621350732091595593omplex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_814
tff(fact_11012_ATP_Olambda__815,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_jq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_815
tff(fact_11013_ATP_Olambda__816,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_ji(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_816
tff(fact_11014_ATP_Olambda__817,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ju(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_817
tff(fact_11015_ATP_Olambda__818,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_jk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_818
tff(fact_11016_ATP_Olambda__819,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_jw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_819
tff(fact_11017_ATP_Olambda__820,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_jm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_820
tff(fact_11018_ATP_Olambda__821,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_jy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_821
tff(fact_11019_ATP_Olambda__822,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: bool] : aa_bool_assn(aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_bm(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uuc),aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu),Uub)))) ).

% ATP.lambda_822
tff(fact_11020_ATP_Olambda__823,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_bo(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),vEBT_vebt_succ(Uu,Uub)))) ).

% ATP.lambda_823
tff(fact_11021_ATP_Olambda__824,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_bn(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),vEBT_vebt_pred(Uu,Uub)))) ).

% ATP.lambda_824
tff(fact_11022_ATP_Olambda__825,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_dp(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uub),Uua)),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Uuc),nth_VEBT_VEBTi(Uua,Uu)))) ).

% ATP.lambda_825
tff(fact_11023_ATP_Olambda__826,axiom,
    ! [Uu: nat,Uua: list_option_nat,Uub: array_option_nat,Uuc: option_nat] : aa_option_nat_assn(aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_dr(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_option_nat(Uub,Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),nth_option_nat(Uua,Uu)))) ).

% ATP.lambda_826
tff(fact_11024_ATP_Olambda__827,axiom,
    ! [Uu: nat,Uua: list_bool,Uub: array_bool,Uuc: bool] : aa_bool_assn(aa_arr3483125744708554596l_assn(aa_lis2577335236791371354l_assn(aTP_Lamm_dq(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_bool(Uub,Uua)),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uuc),aa_nat_bool(nth_bool(Uua),Uu)))) ).

% ATP.lambda_827
tff(fact_11025_ATP_Olambda__828,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat] : aa_nat_assn(aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ds(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_nat(Uub,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),aa_nat_nat(nth_nat(Uua),Uu)))) ).

% ATP.lambda_828
tff(fact_11026_ATP_Olambda__829,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_afi(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(Uu,Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_829
tff(fact_11027_ATP_Olambda__830,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_afg(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_830
tff(fact_11028_ATP_Olambda__831,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_afo(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_831
tff(fact_11029_ATP_Olambda__832,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_afk(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_832
tff(fact_11030_ATP_Olambda__833,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_afm(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_833
tff(fact_11031_ATP_Olambda__834,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_fun1713859001248219487nteger(aTP_Lamm_adm(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(Uu,Uuc)),aa_nat_Code_integer(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_834
tff(fact_11032_ATP_Olambda__835,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_adk(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_835
tff(fact_11033_ATP_Olambda__836,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ads(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_836
tff(fact_11034_ATP_Olambda__837,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ado(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_837
tff(fact_11035_ATP_Olambda__838,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_adv(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_838
tff(fact_11036_ATP_Olambda__839,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_adq(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_839
tff(fact_11037_ATP_Olambda__840,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_nr(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_840
tff(fact_11038_ATP_Olambda__841,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_nt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_841
tff(fact_11039_ATP_Olambda__842,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_np(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_842
tff(fact_11040_ATP_Olambda__843,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_nv(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_843
tff(fact_11041_ATP_Olambda__844,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_zq(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_zp(Uu),Uub),Uuc))) ).

% ATP.lambda_844
tff(fact_11042_ATP_Olambda__845,axiom,
    ! [Uu: set_int,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8197081679332685312t_bool,Uuc: code_integer] : aa_Code_integer_int(aa_fun531554186711482826er_int(aa_fun2196751070425669740er_int(aTP_Lamm_zl(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_Cod5491795744640130032nt_int(Uua,Uuc),collect_int(aa_Cod8597349444926882409t_bool(aa_fun6946700965406361788t_bool(aTP_Lamm_zk(Uu),Uub),Uuc))) ).

% ATP.lambda_845
tff(fact_11043_ATP_Olambda__846,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_zh(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_zg(Uu),Uub),Uuc))) ).

% ATP.lambda_846
tff(fact_11044_ATP_Olambda__847,axiom,
    ! [Uu: set_int,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_zd(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_VEB7166967446021329869nt_int(Uua,Uuc),collect_int(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_zc(Uu),Uub),Uuc))) ).

% ATP.lambda_847
tff(fact_11045_ATP_Olambda__848,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_yz(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_mc(Uu),Uub),Uuc))) ).

% ATP.lambda_848
tff(fact_11046_ATP_Olambda__849,axiom,
    ! [Uu: set_nat,Uua: fun_Co822469698833053651at_int,Uub: fun_Co8657364630979331932t_bool,Uuc: code_integer] : aa_Code_integer_int(aa_fun7994889559813147430er_int(aa_fun6206124202175859492er_int(aTP_Lamm_yw(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_Cod4492442032739075476at_int(Uua,Uuc),collect_nat(aa_Cod2498268374540730821t_bool(aa_fun498830667440607164t_bool(aTP_Lamm_lx(Uu),Uub),Uuc))) ).

% ATP.lambda_849
tff(fact_11047_ATP_Olambda__850,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_zw(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_lo(Uu),Uub),Uuc))) ).

% ATP.lambda_850
tff(fact_11048_ATP_Olambda__851,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_yt(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_lo(Uu),Uub),Uuc))) ).

% ATP.lambda_851
tff(fact_11049_ATP_Olambda__852,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_zu(Uu),Uua),Uub),Uuc) = groups708209901874060359at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_lk(Uu),Uub),Uuc))) ).

% ATP.lambda_852
tff(fact_11050_ATP_Olambda__853,axiom,
    ! [Uu: set_nat,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun6141810409722670662BT_int(aa_fun6474440919679335769BT_int(aTP_Lamm_yq(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_VEB6167613734120275313at_int(Uua,Uuc),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_lk(Uu),Uub),Uuc))) ).

% ATP.lambda_853
tff(fact_11051_ATP_Olambda__854,axiom,
    ! [Uu: set_complex,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co7124838579184476414x_bool,Uuc: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7983975303565649482omplex(aa_fun843835629537646824omplex(aTP_Lamm_na(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_Cod3397930797123873268omplex(Uua,Uuc),collect_complex(aa_Cod6819470024986797159x_bool(aa_fun3619973020723243580x_bool(aTP_Lamm_mz(Uu),Uub),Uuc))) ).

% ATP.lambda_854
tff(fact_11052_ATP_Olambda__855,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: int] : aa_int_complex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_mu(Uu),Uub),Uuc))) ).

% ATP.lambda_855
tff(fact_11053_ATP_Olambda__856,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_mq(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_mp(Uu),Uub),Uuc))) ).

% ATP.lambda_856
tff(fact_11054_ATP_Olambda__857,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_mm(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ml(Uu),Uub),Uuc))) ).

% ATP.lambda_857
tff(fact_11055_ATP_Olambda__858,axiom,
    ! [Uu: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_mi(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_mh(Uu),Uub),Uuc))) ).

% ATP.lambda_858
tff(fact_11056_ATP_Olambda__859,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_md(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_mc(Uu),Uub),Uuc))) ).

% ATP.lambda_859
tff(fact_11057_ATP_Olambda__860,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co8657364630979331932t_bool,Uuc: code_integer] : aa_Code_integer_nat(aa_fun2949368542467568330er_nat(aa_fun8366171557965857828er_nat(aTP_Lamm_ly(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod2498268374540730821t_bool(aa_fun498830667440607164t_bool(aTP_Lamm_lx(Uu),Uub),Uuc))) ).

% ATP.lambda_860
tff(fact_11058_ATP_Olambda__861,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_lt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ls(Uu),Uub),Uuc))) ).

% ATP.lambda_861
tff(fact_11059_ATP_Olambda__862,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_lp(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_lo(Uu),Uub),Uuc))) ).

% ATP.lambda_862
tff(fact_11060_ATP_Olambda__863,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_ll(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_lk(Uu),Uub),Uuc))) ).

% ATP.lambda_863
tff(fact_11061_ATP_Olambda__864,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bu(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_864
tff(fact_11062_ATP_Olambda__865,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bt(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_865
tff(fact_11063_ATP_Olambda__866,axiom,
    ! [Uu: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_866
tff(fact_11064_ATP_Olambda__867,axiom,
    ! [Uu: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: option_nat] : aa_option_nat_assn(aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_fc(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu),aa_option_nat_assn(aa_VEB931861714549014589t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_867
tff(fact_11065_ATP_Olambda__868,axiom,
    ! [Uu: assn,Uua: fun_VE1323945509527881313l_assn,Uub: vEBT_VEBT,Uuc: bool] : aa_bool_assn(aa_VEB2879522196611654106l_assn(aa_fun6755658226996116224l_assn(aTP_Lamm_fa(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu),aa_bool_assn(aa_VEB2879522196611654106l_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_868
tff(fact_11066_ATP_Olambda__869,axiom,
    ! [Uu: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat] : aa_nat_assn(aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_fe(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu),aa_nat_assn(aa_VEB8640158415945897581t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_869
tff(fact_11067_ATP_Olambda__870,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_qq(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_870
tff(fact_11068_ATP_Olambda__871,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qp(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_871
tff(fact_11069_ATP_Olambda__872,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abk(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_872
tff(fact_11070_ATP_Olambda__873,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_pz(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_873
tff(fact_11071_ATP_Olambda__874,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_874
tff(fact_11072_ATP_Olambda__875,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_abj(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_875
tff(fact_11073_ATP_Olambda__876,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_air(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_876
tff(fact_11074_ATP_Olambda__877,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ait(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_877
tff(fact_11075_ATP_Olambda__878,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aiv(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_878
tff(fact_11076_ATP_Olambda__879,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aix(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_879
tff(fact_11077_ATP_Olambda__880,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_aff(Uu),Uua),Uub),Uuc),Uud) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_afe(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_880
tff(fact_11078_ATP_Olambda__881,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_afd(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_afc(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_881
tff(fact_11079_ATP_Olambda__882,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_afb(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_afa(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_882
tff(fact_11080_ATP_Olambda__883,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: code_integer,Uuc: code_integer,Uud: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aa_fun4629701700493761862nteger(aTP_Lamm_aez(Uu),Uua),Uub),Uuc),Uud) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aa_Cod8866121339495654636nteger(aTP_Lamm_aey(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_883
tff(fact_11081_ATP_Olambda__884,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_aex(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_aew(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_884
tff(fact_11082_ATP_Olambda__885,axiom,
    ! [Uu: fun_VE1571388752190420990T_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis7580683478039494528i_assn(aTP_Lamm_cx(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L1279224858307276611T_VEBT(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_885
tff(fact_11083_ATP_Olambda__886,axiom,
    ! [Uu: fun_VE6910866152645720856l_assn,Uua: list_VEBT_VEBT,Uub: list_real,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis1734048899804268331i_assn(aa_lis578560538171925480i_assn(aTP_Lamm_da(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L5781919052683127133T_real(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_886
tff(fact_11084_ATP_Olambda__887,axiom,
    ! [Uu: fun_VE1323945509527881313l_assn,Uua: list_VEBT_VEBT,Uub: list_bool,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis3429412498766249874i_assn(aa_lis8505491871031121891i_assn(aTP_Lamm_cz(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(aa_list_bool_assn(vEBT_L5320321625355440422T_bool(Uu,Uua),Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_887
tff(fact_11085_ATP_Olambda__888,axiom,
    ! [Uu: fun_VE4931415397359990460t_assn,Uua: list_VEBT_VEBT,Uub: list_nat,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis3335532474104912652i_assn(aTP_Lamm_cy(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uu,Uua),Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_888
tff(fact_11086_ATP_Olambda__889,axiom,
    ! [Uu: fun_re560840401387337987i_assn,Uua: list_real,Uub: list_VEBT_VEBTi,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis6960512757483956916i_assn(aa_lis4969729103256914541i_assn(aTP_Lamm_dc(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L9060850011106065574_VEBTi(Uu,Uua),Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_889
tff(fact_11087_ATP_Olambda__890,axiom,
    ! [Uu: fun_re8050108611700907676T_assn,Uua: list_real,Uub: list_VEBT_VEBT,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis9172383540912000132i_assn(aTP_Lamm_db(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L4595930785310033027T_VEBT(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_890
tff(fact_11088_ATP_Olambda__891,axiom,
    ! [Uu: fun_re6303261851654292474l_assn,Uua: list_real,Uub: list_real,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis1734048899804268331i_assn(aa_lis8605243017695430628i_assn(aTP_Lamm_df(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L1930518968523514909l_real(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_891
tff(fact_11089_ATP_Olambda__892,axiom,
    ! [Uu: fun_re2206215847901373695l_assn,Uua: list_real,Uub: list_bool,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis3429412498766249874i_assn(aa_lis5807736783900374247i_assn(aTP_Lamm_de(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L870199270341830246l_bool(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_892
tff(fact_11090_ATP_Olambda__893,axiom,
    ! [Uu: fun_re574865750389657758t_assn,Uua: list_real,Uub: list_nat,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis4680633809818909960i_assn(aTP_Lamm_dd(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L1446010312343316929al_nat(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_893
tff(fact_11091_ATP_Olambda__894,axiom,
    ! [Uu: fun_bo5394993412183903963T_assn,Uua: list_bool,Uub: list_VEBT_VEBT,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis6766312878945579869i_assn(aTP_Lamm_dg(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L2136499394512906784T_VEBT(Uu,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ).

% ATP.lambda_894
tff(fact_11092_ATP_Olambda__895,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ajl(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ajk(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_895
tff(fact_11093_ATP_Olambda__896,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_afp(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_afo(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud)) ).

% ATP.lambda_896
tff(fact_11094_ATP_Olambda__897,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_afn(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_afm(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),Uud)) ).

% ATP.lambda_897
tff(fact_11095_ATP_Olambda__898,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_afl(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_afk(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_898
tff(fact_11096_ATP_Olambda__899,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: code_integer,Uuc: code_integer,Uud: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aa_fun4629701700493761862nteger(aTP_Lamm_afj(Uu),Uua),Uub),Uuc),Uud) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_afi(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),Uud)) ).

% ATP.lambda_899
tff(fact_11097_ATP_Olambda__900,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_afh(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_afg(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud)) ).

% ATP.lambda_900
tff(fact_11098_ATP_Olambda__901,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_aer(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_901
tff(fact_11099_ATP_Olambda__902,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_aev(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_902
tff(fact_11100_ATP_Olambda__903,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_aet(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_903
tff(fact_11101_ATP_Olambda__904,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_aeq(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_904
tff(fact_11102_ATP_Olambda__905,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_aeu(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_905
tff(fact_11103_ATP_Olambda__906,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_aes(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_906
tff(fact_11104_ATP_Olambda__907,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: code_integer,Uuc: nat,Uud: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aa_Cod8866121339495654636nteger(aTP_Lamm_aey(Uu),Uua),Uub),Uuc),Uud) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),Uud))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_907
tff(fact_11105_ATP_Olambda__908,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_aew(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uuc)) ).

% ATP.lambda_908
tff(fact_11106_ATP_Olambda__909,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_afe(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_909
tff(fact_11107_ATP_Olambda__910,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_afa(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_910
tff(fact_11108_ATP_Olambda__911,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_afc(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),Uud))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_911
tff(fact_11109_ATP_Olambda__912,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: array_VEBT_VEBTi] : aa_arr8637723718858317150i_assn(aa_VEB7841561841333234298i_assn(aa_arr6331738488200487899i_assn(aa_lis1967528460011584869i_assn(aTP_Lamm_agv(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uub),list_u6098035379799741383_VEBTi(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal8089995073349378887_VEBTi(Uud,Uub))) ).

% ATP.lambda_912
tff(fact_11110_ATP_Olambda__913,axiom,
    ! [Uu: nat,Uua: list_bool,Uub: array_bool,Uuc: bool,Uud: array_bool] : aa_array_bool_assn(aa_boo8717665131543412656l_assn(aa_arr1850246612958046993l_assn(aa_lis1773218934604694151l_assn(aTP_Lamm_agt(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(snga_assn_bool(Uub,list_update_bool(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal_array_bool(Uud,Uub))) ).

% ATP.lambda_913
tff(fact_11111_ATP_Olambda__914,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat,Uud: array_nat] : aa_array_nat_assn(aa_nat8392338049721622586t_assn(aa_arr4802002299793635008t_assn(aa_lis5007279772662053221t_assn(aTP_Lamm_agu(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(snga_assn_nat(Uub,list_update_nat(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal_array_nat(Uud,Uub))) ).

% ATP.lambda_914
tff(fact_11112_ATP_Olambda__915,axiom,
    ! [Uu: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_ez(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_915
tff(fact_11113_ATP_Olambda__916,axiom,
    ! [Uu: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_option_nat] : aa_lis1398067190886521878t_assn(aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_fd(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu),aa_lis1398067190886521878t_assn(vEBT_L8010285020845282001on_nat(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_916
tff(fact_11114_ATP_Olambda__917,axiom,
    ! [Uu: assn,Uua: fun_VE1323945509527881313l_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_bool] : aa_list_bool_assn(aa_nat2784677144853144906l_assn(aa_VEB3184745993655927913l_assn(aa_fun7303525306808986255l_assn(aTP_Lamm_fb(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu),aa_list_bool_assn(vEBT_L5320321625355440422T_bool(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_917
tff(fact_11115_ATP_Olambda__918,axiom,
    ! [Uu: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_nat] : aa_list_nat_assn(aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_ff(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu),aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_918
tff(fact_11116_ATP_Olambda__919,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi] : aa_VEB4941942288398148609T_VEBT(aTP_Lamm_gi(Uu),Uua) = Uu ).

% ATP.lambda_919
tff(fact_11117_ATP_Olambda__920,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_ge(Uu),Uua) = Uu ).

% ATP.lambda_920
tff(fact_11118_ATP_Olambda__921,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_real_VEBT_VEBT(aTP_Lamm_gf(Uu),Uua) = Uu ).

% ATP.lambda_921
tff(fact_11119_ATP_Olambda__922,axiom,
    ! [Uu: vEBT_VEBT,Uua: bool] : aa_bool_VEBT_VEBT(aTP_Lamm_gg(Uu),Uua) = Uu ).

% ATP.lambda_922
tff(fact_11120_ATP_Olambda__923,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_gh(Uu),Uua) = Uu ).

% ATP.lambda_923
tff(fact_11121_ATP_Olambda__924,axiom,
    ! [Uu: assn,Uua: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aTP_Lamm_ef(Uu),Uua) = Uu ).

% ATP.lambda_924
tff(fact_11122_ATP_Olambda__925,axiom,
    ! [Uu: real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_gd(Uu),Uua) = Uu ).

% ATP.lambda_925
tff(fact_11123_ATP_Olambda__926,axiom,
    ! [Uu: bool,Uua: product_unit] :
      ( pp(aa_Product_unit_bool(aTP_Lamm_ua(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_926
tff(fact_11124_ATP_Olambda__927,axiom,
    ! [Uu: bool,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ty(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_927
tff(fact_11125_ATP_Olambda__928,axiom,
    ! [Uu: bool,Uua: numeral_num1] :
      ( pp(aa_Numeral_num1_bool(aTP_Lamm_ub(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_928
tff(fact_11126_ATP_Olambda__929,axiom,
    ! [Uu: bool,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_tx(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_929
tff(fact_11127_ATP_Olambda__930,axiom,
    ! [Uu: bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_uc(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_930
tff(fact_11128_ATP_Olambda__931,axiom,
    ! [Uu: bool,Uua: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_ud(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_931
tff(fact_11129_ATP_Olambda__932,axiom,
    ! [Uu: bool,Uua: nat] :
      ( pp(aa_nat_bool(aa_bool_fun_nat_bool(aTP_Lamm_ue,Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_932
tff(fact_11130_ATP_Olambda__933,axiom,
    ! [Uu: bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_tz(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_933
tff(fact_11131_ATP_Olambda__934,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_nat(aTP_Lamm_gc(Uu),Uua) = Uu ).

% ATP.lambda_934
tff(fact_11132_ATP_Olambda__935,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_sj,Uu) = Uu ).

% ATP.lambda_935
tff(fact_11133_ATP_Olambda__936,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_af,Uu) = Uu ).

% ATP.lambda_936
tff(fact_11134_ATP_Olambda__937,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ag,Uu) = Uu ).

% ATP.lambda_937
tff(fact_11135_ATP_Olambda__938,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ah,Uu) = Uu ).

% ATP.lambda_938
tff(fact_11136_ATP_Olambda__939,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ai,Uu) = Uu ).

% ATP.lambda_939
tff(fact_11137_ATP_Olambda__940,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aj,Uu) = Uu ).

% ATP.lambda_940
tff(fact_11138_ATP_Olambda__941,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bg,Uu) = zero_zero_complex ).

% ATP.lambda_941
tff(fact_11139_ATP_Olambda__942,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bh,Uu) = zero_zero_real ).

% ATP.lambda_942
tff(fact_11140_ATP_Olambda__943,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_bi,Uu) = zero_zero_rat ).

% ATP.lambda_943
tff(fact_11141_ATP_Olambda__944,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_ur,Uu) = zero_zero_complex ).

% ATP.lambda_944
tff(fact_11142_ATP_Olambda__945,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jz,Uu) = zero_zero_real ).

% ATP.lambda_945
tff(fact_11143_ATP_Olambda__946,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bj,Uu) = zero_zero_nat ).

% ATP.lambda_946
tff(fact_11144_ATP_Olambda__947,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_us,Uu) = zero_zero_int ).

% ATP.lambda_947
tff(fact_11145_ATP_Olambda__948,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_bk,Uu) = zero_zero_int ).

% ATP.lambda_948
tff(fact_11146_ATP_Olambda__949,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_xd,Uu) = one_one_nat ).

% ATP.lambda_949
tff(fact_11147_ATP_Olambda__950,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_xb,Uu) = one_one_int ).

% ATP.lambda_950
tff(fact_11148_ATP_Olambda__951,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_xc,Uu) = one_one_int ).

% ATP.lambda_951
tff(fact_11149_ATP_Olambda__952,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ar,Uu))
    <=> $false ) ).

% ATP.lambda_952
tff(fact_11150_ATP_Olambda__953,axiom,
    ! [Uu: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_aq,Uu))
    <=> $false ) ).

% ATP.lambda_953
tff(fact_11151_ATP_Olambda__954,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_ahx,Uu))
    <=> $false ) ).

% ATP.lambda_954
tff(fact_11152_ATP_Olambda__955,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_as,Uu))
    <=> $false ) ).

% ATP.lambda_955
tff(fact_11153_ATP_Olambda__956,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_at,Uu))
    <=> $false ) ).

% ATP.lambda_956
tff(fact_11154_ATP_Olambda__957,axiom,
    ! [Uu: product_unit] :
      ( pp(aa_Product_unit_bool(aTP_Lamm_uy,Uu))
    <=> $true ) ).

% ATP.lambda_957
tff(fact_11155_ATP_Olambda__958,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ux,Uu))
    <=> $true ) ).

% ATP.lambda_958
tff(fact_11156_ATP_Olambda__959,axiom,
    ! [Uu: numeral_num1] :
      ( pp(aa_Numeral_num1_bool(aTP_Lamm_uz,Uu))
    <=> $true ) ).

% ATP.lambda_959
tff(fact_11157_ATP_Olambda__960,axiom,
    ! [Uu: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_uw,Uu))
    <=> $true ) ).

% ATP.lambda_960
tff(fact_11158_ATP_Olambda__961,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_va,Uu))
    <=> $true ) ).

% ATP.lambda_961
tff(fact_11159_ATP_Olambda__962,axiom,
    ! [Uu: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_vb,Uu))
    <=> $true ) ).

% ATP.lambda_962
tff(fact_11160_ATP_Olambda__963,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_vc,Uu))
    <=> $true ) ).

% ATP.lambda_963
tff(fact_11161_ATP_Olambda__964,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_uv,Uu))
    <=> $true ) ).

% ATP.lambda_964
tff(fact_11162_ATP_Olambda__965,axiom,
    ! [Uu: vEBT_VEBTi] : aa_VEB205140289345309584at_nat(aTP_Lamm_dn,Uu) = suc ).

% ATP.lambda_965
tff(fact_11163_ATP_Olambda__966,axiom,
    ! [Uu: vEBT_VEBT] : aa_VEB1122092716774696213at_nat(aTP_Lamm_dj,Uu) = suc ).

% ATP.lambda_966
tff(fact_11164_ATP_Olambda__967,axiom,
    ! [Uu: real] : aa_real_fun_nat_nat(aTP_Lamm_dk,Uu) = suc ).

% ATP.lambda_967
tff(fact_11165_ATP_Olambda__968,axiom,
    ! [Uu: bool] : aa_bool_fun_nat_nat(aTP_Lamm_dl,Uu) = suc ).

% ATP.lambda_968
tff(fact_11166_ATP_Olambda__969,axiom,
    ! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_dm,Uu) = suc ).

% ATP.lambda_969

% Helper facts (77)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fimplies(P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fimplies(P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fimplies(P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y3: int] : aa_int_int(aa_int_fun_int_int(if_int(fFalse),X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y3: int] : aa_int_int(aa_int_fun_int_int(if_int(fTrue),X2),Y3) = X2 ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X2),Y3) = X2 ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y3: rat] : aa_rat_rat(if_rat(fFalse,X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y3: rat] : aa_rat_rat(if_rat(fTrue,X2),Y3) = X2 ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y3: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X2),Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X2: int,Y3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X2: bool,Y3: bool] :
      ( ( X2 != Y3 )
      | pp(aa_bool_bool(fequal_bool(X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X2: bool,Y3: bool] :
      ( ~ pp(aa_bool_bool(fequal_bool(X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).

tff(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X2: assn,Y3: assn] : if_assn(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X2: assn,Y3: assn] : if_assn(fTrue,X2,Y3) = X2 ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(if_complex(fFalse),X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(if_complex(fTrue),X2),Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y3: complex] :
      ( ( X2 != Y3 )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y3: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y3: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fFalse,X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y3: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fTrue,X2),Y3) = X2 ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fFalse,X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fTrue,X2),Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( X2 != Y3 )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( X2 != Y3 )
      | pp(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y3: option_nat] : if_option_nat(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y3: option_nat] : if_option_nat(fTrue,X2,Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__Nat__Onat_J_T,axiom,
    ! [X2: array_nat,Y3: array_nat] :
      ( ( X2 != Y3 )
      | pp(fequal_array_nat(X2,Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__Nat__Onat_J_T,axiom,
    ! [X2: array_nat,Y3: array_nat] :
      ( ~ pp(fequal_array_nat(X2,Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__HOL__Obool_J_T,axiom,
    ! [X2: array_bool,Y3: array_bool] :
      ( ( X2 != Y3 )
      | pp(fequal_array_bool(X2,Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__HOL__Obool_J_T,axiom,
    ! [X2: array_bool,Y3: array_bool] :
      ( ~ pp(fequal_array_bool(X2,Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X2: vEBT_VEBTi,Y3: vEBT_VEBTi] :
      ( ( X2 != Y3 )
      | pp(aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X2: vEBT_VEBTi,Y3: vEBT_VEBTi] :
      ( ~ pp(aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y3: option_nat] :
      ( ( X2 != Y3 )
      | pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y3: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X2: heap_Time_Heap_nat,Y3: heap_Time_Heap_nat] : if_Hea2662716070787841314ap_nat(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X2: heap_Time_Heap_nat,Y3: heap_Time_Heap_nat] : if_Hea2662716070787841314ap_nat(fTrue,X2,Y3) = X2 ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__HOL__Obool_J_T,axiom,
    ! [X2: heap_Time_Heap_bool,Y3: heap_Time_Heap_bool] : if_Hea7696252282596813915p_bool(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__HOL__Obool_J_T,axiom,
    ! [X2: heap_Time_Heap_bool,Y3: heap_Time_Heap_bool] : if_Hea7696252282596813915p_bool(fTrue,X2,Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: list_VEBT_VEBTi,Y3: list_VEBT_VEBTi] :
      ( ( X2 != Y3 )
      | pp(aa_lis8521252006704567508i_bool(fequal1389750275112723248_VEBTi(X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: list_VEBT_VEBTi,Y3: list_VEBT_VEBTi] :
      ( ~ pp(aa_lis8521252006704567508i_bool(fequal1389750275112723248_VEBTi(X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y3: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y3: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X2,Y3) = X2 ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X2),Y3) = Y3 ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X2),Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: array_VEBT_VEBTi,Y3: array_VEBT_VEBTi] :
      ( ( X2 != Y3 )
      | pp(fequal8089995073349378887_VEBTi(X2,Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: array_VEBT_VEBTi,Y3: array_VEBT_VEBTi] :
      ( ~ pp(fequal8089995073349378887_VEBTi(X2,Y3))
      | ( X2 = Y3 ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( X2 != Y3 )
      | pp(aa_Pro4501881432599481050t_bool(fequal6010348034681699726nt_int(X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
      ( ~ pp(aa_Pro4501881432599481050t_bool(fequal6010348034681699726nt_int(X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,Y3: heap_T8145700208782473153_VEBTi] : if_Hea8453224502484754311_VEBTi(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,Y3: heap_T8145700208782473153_VEBTi] : if_Hea8453224502484754311_VEBTi(fTrue,X2,Y3) = X2 ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X2: heap_T2636463487746394924on_nat,Y3: heap_T2636463487746394924on_nat] : if_Hea5867803462524415986on_nat(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X2: heap_T2636463487746394924on_nat,Y3: heap_T2636463487746394924on_nat] : if_Hea5867803462524415986on_nat(fTrue,X2,Y3) = X2 ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( X2 != Y3 )
      | pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,X2),Y3)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ~ pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,X2),Y3))
      | ( X2 = Y3 ) ) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X2,Y3) = Y3 ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X2,Y3) = X2 ).

% Conjectures (10)
tff(conj_0,hypothesis,
    tia = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),x13),x14) ).

tff(conj_1,hypothesis,
    size_s7982070591426661849_VEBTi(tree_is) = size_s6755466524823107622T_VEBT(treeList) ).

tff(conj_2,hypothesis,
    xa != mi ).

tff(conj_3,hypothesis,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).

tff(conj_4,hypothesis,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),xa)) ).

tff(conj_5,hypothesis,
    xaa = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

tff(conj_6,hypothesis,
    vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) = none_nat ).

tff(conj_7,hypothesis,
    xba = none_nat ).

tff(conj_8,hypothesis,
    xa != ma ).

tff(conj_9,conjecture,
    pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(x13),list_u6098035379799741383_VEBTi(tree_is,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),xb)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),list_u6098035379799741383_VEBTi(tree_is,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),xb)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(treeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary))),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),x13),xc)))) ).

%------------------------------------------------------------------------------